Magnetoelastic standing waves induced in UO2 by microsecond magnetic field pulses

Magnetoelastic dilatometry of the piezomagnetic antiferromagnet UO₂ was performed via the fiber Bragg grating method in magnetic fields up to 150 T generated by a single-turn coil setup. We show that in microsecond timescales, pulsed-magnetic fields excite mechanical resonances at temperatures ranging from 10 to 300 K, in the paramagnetic as well as within the robust antiferromagnetic state of the material. These resonances, which are barely attenuated within the 100-µs observation window, are attributed to the strong magnetoelastic coupling in UO₂ combined with the high crystalline quality of the single crystal samples. They compare well with mechanical resonances obtained by a resonant ultrasound technique and superimpose on the known nonmonotonic magnetostriction background. A clear phase shift of π in the lattice oscillations is observed in the antiferromagnetic state when the magnetic field overcomes the piezomagnetic switch field Hc=−18 T. We present a theoretical argument that explains this unexpected behavior as a result of the reversal of the antiferromagnetic order parameter at Hc.

The antiferromagnetic (AFM) insulator uranium dioxide UO₂ has been the subject of extensive research during the last decades predominantly due to its widespread use as nuclear fuel in commercial power reactors (1). Besides efforts to understand the unusually poor thermal conductivity of UO₂, which impacts its performance as nuclear fuel (2), a recent magnetostriction study in pulsed magnetic fields up to 92 T uncovered linear magnetostriction in UO₂ (3), a hallmark of piezomagnetism.Piezomagnetism is characterized by the induction of a magnetic polarization by application of mechanical strain, which, in the case of UO₂, is enabled by broken time-reversal symmetry in the 3-k AFM structure that emerges below TN=30.8 K (4–7) and is accompanied by a Jahn–Teller distortion of the oxygen cage (8–11). This also leads to a complex hysteretic magnetoelastic memory behavior where magnetic domain switching occurs at fields around ±18 T at T=2.5 K. Interestingly, the very large applied magnetic fields proved unable to suppress the AFM state that sets in at TN (3). These earlier results provide direct evidence for the unusually high energy scale of spin-lattice interactions and call for further studies in higher magnetic fields.Here we present axial magnetostriction data obtained in a UO₂ single crystal in magnetic fields to 150 T. These ultrahigh fields were produced by single-turn coil pulsed resistive magnets (12, 13) and applied along the [111] crystallographic axis at various temperatures between 10 K and room temperature. At all temperatures, we observe a dominant negative magnetostriction proportional to H² accompanied by unexpectedly strong oscillations that establish a mechanical resonance in the sample virtually instantly upon delivery of the 102 T/μs pulsed magnetic field rate of change. The oscillations are long-lasting due to very low losses and match mechanical resonances obtained with a resonant ultrasound spectroscopy (RUS) technique (14). Mechanical resonances were suggested to explain anomalies in magnetostriction measurements during single-turn pulses (15, 16); however, their potential to elucidate magnetic dynamics was not explored so far. When the sample is cooled below room temperature, the frequencies soften, consistent with observations in studies of the UO₂ elastic constant c₄₄ as a function of temperature (17, 18).In the AFM state, i.e., T<30.8 K, the characteristic magnetic field sign switch in our single-turn coil magnet (a feature of destructive magnets) results in applied field values in excess of the UO₂ AFM domain switch field of Hc≈−18 T. This field sign switch exposes yet another unexpected result, namely, a π (180°) phase shift in the magnetoelastic oscillations. We use a driven harmonic oscillator and an analytical model to shed light on the origin of the observed phase shift.     

Modeling the instantaneous normal mode spectra of liquids as that of unstable elastic media

We study the instantaneous normal mode (INM) spectrum of a simulated soft-sphere liquid at different equilibrium temperatures T. We find that the spectrum of eigenvalues ρ(λ) has a sharp maximum near (but not at) λ=0 and decreases monotonically with |λ| on both the stable and unstable sides of the spectrum. The spectral shape strongly depends on temperature. It is rather asymmetric at low temperatures (close to the dynamical critical temperature) and becomes symmetric at high temperatures. To explain these findings we present a mean-field theory for ρ(λ), which is based on a heterogeneous elasticity model, in which the local shear moduli exhibit spatial fluctuations, including negative values. We find good agreement between the simulation data and the model calculations, done with the help of the self-consistent Born approximation (SCBA), when we take the variance of the fluctuations to be proportional to the temperature T. More importantly, we find an empirical correlation of the positions of the maxima of ρ(λ) with the low-frequency exponent of the density of the vibrational modes of the glasses obtained by quenching to T=0 from the temperature T. We discuss the present findings in connection to the liquid to glass transformation and its precursor phenomena.

The investigation of the potential energy surface (PES) V(r1(t)…rN(t)) of a liquid (made up of N particles with positions r1(t)…rN(t) at a time instant t) and the corresponding instantaneous normal modes (INMs) of the (Hessian) matrix of curvatures has been a focus of liquid and glass science since the appearance of Goldstein’s seminal article (1) on the relation between the PES and the liquid dynamics in the viscous regime above the glass transition (2–27).The PES has been shown to form a rather ragged landscape in configuration space (8, 28, 29) characterized by its stationary points. In a glass these points are minima and are called “inherent structures.” The PES is believed to contain important information on the liquid–glass transformation mechanism. For the latter a complete understanding is still missing (28, 30, 31). The existing molecular theory of the liquid–glass transformation is mode-coupling theory (MCT) (32, 33) and its mean-field Potts spin version (28, 34). MCT predicts a sharp transition at a temperature TMCT>Tg, where T_g is the temperature of structural arrest (glass transition temperature). MCT completely misses the heterogeneous activated relaxation processes (dynamical heterogeneities), which are evidently present around and below T_MCT and which are related to the unstable (negative-λ) part of the INM spectrum (28, 30).Near and above T_MCT, apparently, there occurs a fundamental change in the PES. Numerical studies of model liquids have shown that minima present below T_MCT change into saddles, which then explains the absence of activated processes above T_MCT (2–24). Very recently, it was shown that T_MCT is related to a localization–delocalization transition of the unstable INM modes (25, 26).The INM spectrum is obtained in molecular dynamic simulations by diagonalizing the Hessian matrix of the interaction potential, taken at a certain time instant t:Hijαβ(t)=∂2∂xi(α)∂xj(β)V{r1(t)…rN(t)},[1]with ri=(xi(1),xi(2),xi(3)). For large positive values of the eigenvalues λ_j (j=1…N, N being the number of particles in the system) they are related to the square of vibrational frequencies λj=ωj2, and one can consider the Hessian as the counterpart of the dynamical matrix of a solid. In this high-frequency regime one can identify the spectrum with the density of vibrational states (DOS) of the liquid viag(ω)=2 ωρ(λ(ω))=13N∑jδ(ω−ωj) .[2]For small and negative values of λ this identification is not possible. For the unstable part of the spectrum (λ<0) it has become common practice to call the imaginary number λ=iω˜ and define the corresponding DOS asg(ω˜)≡2 ω˜ρ(λ(ω˜)).[3]This function is plotted on the negative ω axis and the stable g(ω), according to [2], on the positive axis. However, the (as we shall see, very interesting) details of the spectrum ρ(λ) near λ = 0 become almost completely hidden by multiplying the spectrum with |ω|. In fact, it has been demonstrated by Sastry et al. (6) and Taraskin and Elliott (7) already 2 decades ago that the INM spectrum of liquids, if plotted as ρ(λ) and not as g(ω) according to [2] and [3], exhibits a characteristic cusp-like maximum at λ = 0. The shape of the spectrum changes strongly with temperature. This is what we find as well in our simulation and what we want to explore further in our present contribution.In the present contribution we demonstrate that the strong change of the spectrum with temperature can be rather well explained in terms of a model, in which the instantaneous harmonic spectrum of the liquid is interpreted to be that of an elastic medium, in which the local shear moduli exhibit strong spatial fluctuations, which includes a large number of negative values. Because these fluctuations are just a snapshot of thermal fluctuations, we assume that they are obeying Gaussian statistics, the variance of which is proportional to the temperature.Evidence for a characteristic change in the liquid configurations in the temperature range above T_g has been obtained in recent simulation studies of the low-frequency vibrational spectrum of glasses, which have been rapidly quenched from a certain parental temperature T*. If T* is decreased from high temperatures toward T_MCT, the low-frequency exponent of the vibrational DOS of the daughter glass (quenched from T* to T = 0) changed from Debye-like g(ω)∝ω2 to g(ω)∝ωs with s > 2. In our numerical investigation of the INM spectra we show a correlation of some details of the low-eigenvalue features of these spectra with the low-frequency properties of the daughter glasses obtained by quenching from the parental temperatures.The stochastic Helmholtz equations (Eq. 7) of an elastic model with spatially fluctuating shear moduli can be readily solved for the averaged Green’s functions by field theoretical techniques (35–37). Via a saddle point approximation with respect to the resulting effective field theory one arrives at a mean-field theory (self-consistent Born approximation [SCBA]) for the self-energy of the averaged Green’s functions. The SCBA predicts a stable spectrum below a threshold value of the variance. Restricted to this stable regime, this theory, called heterogeneous elasticity theory (HET), was rather successful in explaining several low-frequency anomalies in the vibrational spectrum of glasses, including the so-called boson peak, which is an enhancement at finite frequencies over the Debye behavior of the DOS g(ω)∝ω2 (37–41). We now explore the unstable regime of this theory and compare it to the INM spectrum of our simulated soft-sphere liquid.^*We start Results by presenting a comparison of the simulated spectra of the soft-sphere liquid with those obtained by the unstable version of HET-SCBA theory. We then concentrate on some specific features of the INM spectra, namely, the low-eigenvalue slopes and the shift of the spectral maximum from λ = 0. Both features are accounted for by HET-SCBA. In particular, we find an interesting law for the difference between the slopes of the unstable and the stable parts of the spectrum, which behaves as T−2/3, which, again, is accounted for by HET-SCBA.In the end we compare the shift of the spectral maximum with the low-frequency exponent of the DOS of the corresponding daughter glasses and find an empirical correlation. We discuss these results in connection with the saddle to minimum transformation near T_MCT.     

Polar vortex crystals: Emergence and structure

Vortex crystals are quasiregular arrays of like-signed vortices in solid-body rotation embedded within a uniform background of weaker vorticity. Vortex crystals are observed at the poles of Jupiter and in laboratory experiments with magnetized electron plasmas in axisymmetric geometries. We show that vortex crystals form from the free evolution of randomly excited two-dimensional turbulence on an idealized polar cap. Once formed, the crystals are long lived and survive until the end of the simulations (300 crystal-rotation periods). We identify a fundamental length scale, Lγ=(U/γ)1/3, characterizing the size of the crystal in terms of the mean-square velocity U of the fluid and the polar parameter γ=fp/ap2, with f_p the Coriolis parameter at the pole and a_p the polar radius of the planet.

The Juno spacecraft revealed a distinctive form of organized turbulence in the polar atmosphere of Jupiter (1). The North Pole exhibits a central polar cyclone surrounded by eight circumpolar cyclones; the South Pole is characterized by five cyclones surrounding a central polar cyclone. These cyclones are organized into a vortex crystal: a symmetric array of strong vortices in solid-body rotation within a background of weaker vorticity. The Jovian vortex crystals are located within 8° of the poles and have endured since their discovery in 2016 with very little change to their overall structure (2, 3). Jovian crystals are approximately stationary in the System III reference frame (4). The essential physical mechanisms responsible for the emergence and persistence of this peculiar polar dynamical regime are poorly understood.Although observations of polar vortex crystals began with Juno, the most basic model of this phenomenon dates back to Kelvin’s 19th century theory of vortex atoms, i.e., the dynamics of equally spaced point vortices arranged in concentric rings (5–7). Laboratory experiments with magnetized pure-electron plasmas are another antecedent (8–11). This experimental system is isomorphic to a near-ideal two-dimensional (2D) fluid contained within a circular domain, with electron density equivalent to vorticity. Through vortex nucleation, merger, and finally self-organization, regular vortex arrays, in solid-body rotation (9), spontaneously crystallize from a filamented initial vorticity field. These electron-plasma experiments are simpler than polar planetary dynamics in several respects:

• 1)There is no analog of the variation of the Coriolis parameter with latitude (the β-effect);
• 2)The vorticity has strictly one sign (electrons are negatively charged);
• 3)The plasma is contained within a circular domain with a free-slip boundary;
• 4)Plasma dynamics are “barotropic” (the deformation length is infinite); and
• 5)There is effectively no dissipation and there is no energy source (small-scale convection) required to sustain the vortices.

Nonetheless many authors have remarked on the striking resemblance between electron-plasma vortex crystals and Jovian observations (1, 4, 12, 13). While these two systems have many features in common, it is crucial to relax simplifications 1 and 2 in any model of polar atmospheres. In particular, the small variation of the Coriolis parameter in the vicinity of the poles is key.Recent observations show that moist convection drives an upscale energy transfer at Jovian high latitudes (14), consistent with the regime of rapidly rotating Rayleigh–Bénard convection (15). The barotropic component of the flow (i.e., the circumpolar and polar cyclones) exhibits kinetic energy spectra with a k−3 spectral slope, consistent with quasi-geostrophic (QG) dynamics.These results motivate the hypothesis that barotropic QG dynamics, specialized to an idealized “polar-cap” geometry, might explain basic features of the polar planetary regime. Here we show that useful dynamical information, such as the radius of the polar vortex crystal, follows from dimensional analysis of the polar-cap QG dynamics, prepared with an initial condition having a length scale characteristic of convective cells (14).     

Coupled CH4 production and oxidation support CO2 supersaturation in a tropical flood pulse lake (Tonle Sap Lake,Cambodia)

Carbon dioxide (CO₂) supersaturation in lakes and rivers worldwide is commonly attributed to terrestrial–aquatic transfers of organic and inorganic carbon (C) and subsequent, in situ aerobic respiration. Methane (CH₄) production and oxidation also contribute CO₂ to freshwaters, yet this remains largely unquantified. Flood pulse lakes and rivers in the tropics are hypothesized to receive large inputs of dissolved CO₂ and CH₄ from floodplains characterized by hypoxia and reducing conditions. We measured stable C isotopes of CO₂ and CH₄, aerobic respiration, and CH₄ production and oxidation during two flood stages in Tonle Sap Lake (Cambodia) to determine whether dissolved CO₂ in this tropical flood pulse ecosystem has a methanogenic origin. Mean CO₂ supersaturation of 11,000 ± 9,000 μatm could not be explained by aerobic respiration alone. ¹³C depletion of dissolved CO₂ relative to other sources of organic and inorganic C, together with corresponding ¹³C enrichment of CH₄, suggested extensive CH₄ oxidation. A stable isotope-mixing model shows that the oxidation of ¹³C depleted CH₄ to CO₂ contributes between 47 and 67% of dissolved CO₂ in Tonle Sap Lake. ¹³C depletion of dissolved CO₂ was correlated to independently measured rates of CH₄ production and oxidation within the water column and underlying lake sediments. However, mass balance indicates that most of this CH₄ production and oxidation occurs elsewhere, within inundated soils and other floodplain habitats. Seasonal inundation of floodplains is a common feature of tropical freshwaters, where high reported CO₂ supersaturation and atmospheric emissions may be explained in part by coupled CH₄ production and oxidation.

Globally, most lakes and rivers are supersaturated with dissolved carbon dioxide (CO₂) relative to the atmosphere, highlighting their outsized role in transferring and transforming terrestrial carbon (C) (1–3). Terrestrial–aquatic transfers of C can include CO₂ dissolved in terrestrial ground and surface waters (3–6), dissolved inorganic carbon (DIC) from carbonate weathering (7, 8), or organic C from various sources that is subsequently respired in lakes and rivers (9, 10). Initially, oceanic export was thought to be the only fate for terrestrial–aquatic transfers of C, but a growing body of research on sediment burial of organic C and CO₂ emissions from freshwaters prompted the “active pipe” revision to this initial set of assumptions (11). Although freshwaters are now recognized as focal points for transferring and transforming C on the landscape, most of this research has been conducted within temperate freshwaters (2, 11, 12). Few studies focus on the mechanisms of CO₂ supersaturation in tropical lakes and rivers, with most conducted in just one watershed, the Amazon (4, 13–15).CO₂ supersaturation within tropical freshwaters is likely influenced by their unique flood pulse hydrology. The canonical flood pulse concept hypothesizes that annual flooding of riparian land will lead to organic C mobilization and respiration (16). Partial pressures of CO₂ (pCO₂) have been measured in excess of 44,000 μatm in the Amazon River (13), 16,000 μatm in the Congo River (17), and 12,000 μatm in the Lukulu River (17). Richey et al. (13), Borges et al. (18), and Zuidgeest et al. (17) have each shown that that riverine pCO₂ scales with the amount of land flooded in these watersheds. Yet it was only recently that Abril and Borges (19) proposed the importance of flooded land to the “active pipe.” These authors differentiate uplands that unidirectionally drain water downhill (via ground and surface water) from floodplains that bidirectionally exchange water with lakes and rivers (19). They conceptualize how floodplains combine high hydrologic connectivity, high rates of primary production, and high rates of respiration to transfer relatively large amounts of C to tropical freshwaters (19).Methanogenesis inevitably results on floodplains after dissolved oxygen (O₂) and other electron acceptors for anaerobic respiration such as iron and sulfate are consumed (16, 19). Horizontal gradients in dissolved O₂ and reducing conditions have been observed extending from the center of lakes and rivers through their floodplains in the Mekong (20, 21), Congo (22), Pantanal (23), and Amazon watersheds (4). CH₄ production and oxidation occur along such redox gradients (4, 16, 19, 23). CH₄ is produced by acetate fermentation (Eq. 1) and carbonate reduction (Eq. 2) within freshwaters (24, 25). CH₄ production coupled with aerobic oxidation results in CO₂ (Eq. 3 and ref. 25), yet no studies have quantified the relative contribution of coupled CH₄ production and oxidation to CO₂ supersaturation within tropical freshwaters.CH3COOH→CO2+CH4,[1]CO2+8H++8e−→CH4+2H2O,[2]CH4+2O2→CO2+2H2O.[3]The relative contribution of coupled CH₄ production and oxidation to CO₂ supersaturation within tropical freshwaters can be traced with stable C isotopes of CO₂ and CH₄. Methanogenesis results in CH₄ that is depleted in ¹³C (δ¹³C = −65 to −50‰ from acetate fermentation and −110 to −60‰ from carbonate reduction) compared to other potential sources of organic and inorganic C (δ¹³C = −37 to −7.7‰; see Materials and Methods) (24–26). The oxidation of this ¹³C-depleted CH₄ results in ¹³C-depleted CO₂ (24–26). At the same time, CH₄ oxidation enriches the ¹³C/¹²C of residual CH₄ as bacteria and archaea preferentially oxidize ¹²C-CH₄ (25). This means that the ¹³C/¹²C of CO₂ and CH₄ can serve as powerful tools to determine the source of CO₂ supersaturation within freshwaters.Tonle Sap Lake (TSL) is Southeast Asia’s largest lake and an understudied flood pulse ecosystem that supports a regionally important fishery (21, 22, 27). Each May through October, monsoonal rains and Himalayan snowmelt increase discharge in the Mekong River and cause one of its tributaries, the Tonle Sap River, to reverse course from southeast to northwest (21). During this course reversal, the Tonle Sap River floods TSL. The TSL flood pulse increases lake volume from 1.6 to 60 km³ and inundates 12,000 km² of floodplain for 3 to 6 mo per year (21, 27). Holtgrieve et al. (22) have shown that aerobic respiration is consistently greater than primary production in TSL (i.e., net heterotrophy), with the expectation of consistent CO₂ supersaturation. But, the partial pressures, C isotopic compositions, and ultimately the source of dissolved CO₂ in TSL remain unquantified.To quantify CO₂ supersaturation and its origins in TSL, we measured the partial pressures of CO₂ and CH₄ and compared their C isotopic composition to other potential sources of organic and inorganic C. We carried out these measurements in distinct lake environments during the high-water and falling-water stages of the flood pulse, hypothesizing that CH₄ production and oxidation on the TSL floodplain would support CO₂ supersaturation during the high-water stage. We found that coupled CH₄ production and oxidation account for a nontrivial proportion of the total dissolved CO₂ in all TSL environments and during both flood stages, showing that anaerobic degradation of organic C at aquatic–terrestrial transitions can support CO₂ supersaturation within tropical freshwaters.     

On randomly changing conformity bias in cultural transmission

Humans and nonhuman animals display conformist as well as anticonformist biases in cultural transmission. Whereas many previous mathematical models have incorporated constant conformity coefficients, empirical research suggests that the extent of (anti)conformity in populations can change over time. We incorporate stochastic time-varying conformity coefficients into a widely used conformity model, which assumes a fixed number n of “role models” sampled by each individual. We also allow the number of role models to vary over time (nt). Under anticonformity, nonconvergence can occur in deterministic and stochastic models with different parameter values. Even if strong anticonformity may occur, if conformity or random copying (i.e., neither conformity nor anticonformity) is expected, there is convergence to one of the three equilibria seen in previous deterministic models of conformity. Moreover, this result is robust to stochastic variation in nt. However, dynamic properties of these equilibria may be different from those in deterministic models. For example, with random conformity coefficients, all equilibria can be stochastically locally stable simultaneously. Finally, we study the effect of randomly changing weak selection. Allowing the level of conformity, the number of role models, and selection to vary stochastically may produce a more realistic representation of the wide range of group-level properties that can emerge under (anti)conformist biases. This promises to make interpretation of the effect of conformity on differences between populations, for example those connected by migration, rather difficult. Future research incorporating finite population sizes and migration would contribute added realism to these models.

Cavalli-Sforza and Feldman (1) studied the finite population dynamics of a trait whose transmission from one generation to the next depended on the mean value of that trait in the population. This “group transmission” constrained the within-group variability but could lead to increasing variance in the average trait value between groups. Other analyses of cultural transmission biases have incorporated characteristics of trait variation, such as the quality, and characteristics of transmitters, including success and prestige (2). Another class of transmission biases is couched in terms of the frequencies of the cultural variants in the population (3). These “frequency-dependent” biases include conformity and anticonformity, which occur when a more common variant is adopted at a rate greater or less than its population frequency, respectively (4).Humans have exhibited conformity in mental rotation (5), line discrimination (6), and numerical discrimination tasks (7). Anticonformity has been exhibited by young children performing numerical discrimination (7). Unbiased frequency-dependent transmission, known as random copying (8), has been suggested to account for choices of dog breeds (9), Neolithic pottery motifs, patent citations, and baby names (10, 11). However, baby name distributions appear more consistent with frequency-dependent (8, 12) and/or other (13, 14) biases.In nonhuman animals, conformity has been observed in nine-spined sticklebacks choosing a feeder (15) and great tits solving a puzzle box (16, 17) (but see ref. 18). Fruit flies displayed both conformist and anticonformist bias with respect to mate choice (19) (but these authors used a different definition of anticonformity from that of ref. 4, which we use, and therefore did not consider these behaviors to be anticonformist).Asch (20, 21) used a different definition of conformity from ref. 4, namely “the overriding of personal knowledge or behavioral dispositions by countervailing options observed in others” (ref. 22, p. 34). Aschian conformity (22) has been observed in chimpanzees (23, 24), capuchin monkeys (25, 26) (but see ref. 27), vervet monkeys (28), and great tits (16). It has also been empirically tested in at least 133 studies of humans and, in the United States, has declined from the 1950s to the 1990s (29).Temporal variation may also occur in forms of conformity other than Aschian. In ref. 12, popular US baby names from 1960 to 2010 show a concave turnover function indicative of negative frequency-dependent bias, but male baby names from earlier decades (1880 to 1930) show a convex turnover indicative of positive frequency-dependent or direct bias. However, most previous mathematical models of conformity have incorporated constant, rather than time-dependent, conformity coefficients.Cavalli-Sforza and Feldman (ref. 3, chap. 3) and Boyd and Richerson (ref. 4, chap. 7) studied models of frequency-dependent transmission of a cultural trait with two variants. Boyd and Richerson (4) incorporated conformist and anticonformist bias through a conformity coefficient denoted by D. In their simplest model, if the frequency of variant A is p and that of variant B is 1−p, then the frequency of variant A in the offspring generation, p′, isp′=p+Dp(1−p)(2p−1),[1]where D>0 entails conformity (A increases if its frequency is p>12), D<0 entails anticonformity, D=0 entails random copying, and −2<D<1. In this model, each offspring samples the cultural variants of n=3 members of the parental generation (hereafter, role models). Sampling n>3 role models requires different constraints and, if n>4, there are multiple conformity coefficients (Eq. 19).Many subsequent models have built upon Boyd and Richerson’s (4) simplest model (Eq. 1). These have incorporated individual learning, information inaccuracy due to environmental change (30–34), group selection (35), and other transmission biases, including payoff bias (36), direct bias, and prestige bias (37). Other models, which include a single conformity coefficient and preserve the essential features of Eq. 1, incorporate individual learning, environmental variability (32, 38), group selection (39), and multiple cultural variants (38).In agent-based statistical physics models, the up and down spins of an electron are analogous to cultural variants A and B (40, 41). Individuals are nodes in a network and choose among a series of actions with specified probabilities, such as independently acquiring a spin, or sampling neighboring individuals and adopting the majority or minority spin in the sample. The number of sampled role models can be greater than three (42, 43). (Anti)conformity may occur if all (42–47), or if at least r (40, 48), sampled individuals have the same variant. In contrast, Boyd and Richerson’s (4) general model (Eq. 19) allows, for example, stronger conformity to a 60% majority of role models and weaker conformity or anticonformity to a 95% majority (in humans, this might result from a perceived difference between “up-and-coming” and “overly popular” variants).In Boyd and Richerson’s (4) general model, individuals sample n role models, which is more realistic than restricting n to 3 (as in Eq. 1); individuals may be able to observe more than three members of the previous generation. With n>4, different levels of (anti)conformity may occur for different samples j of n role models with one variant. In addition to the example above with 60 and 95% majorities, other relationships between the level of conformity and the sample j of n are possible. For example, the strength of conformity might increase as the number of role models with the more common variant increases. In a recent exploration of Boyd and Richerson’s (4) general model, we found dynamics that departed significantly from those of Eq. 1 (49). If conformity and anticonformity occur for different majorities j of n role models (i.e., j>n2), polymorphic equilibria may exist that were not possible with Eq. 1. In addition, strong enough anticonformity can produce nonconvergence: With as few as 5 role models, stable cycles in variant frequencies may arise, and with as few as 10 role models, chaos is possible. Such complex dynamics may occur with or without selection.Here, we extend both Boyd and Richerson’s (4) simplest (Eq. 1) and general (Eq. 19) models to allow the conformity coefficient(s) to vary randomly across generations, by sampling them from probability distributions. Although some agent-based models allow individuals to switch between “conformist” and “non-” or “anticonformist” states over time (40, 42, 47, 50, 51), to our knowledge, random temporal variation in the conformity coefficients themselves has not been modeled previously. In reality, the degree to which groups of individuals conform may change over time, as illustrated by the finding that young children anti-conformed while older children conformed in a discrimination task (7); thus, it seems reasonable to expect that different generations may also exhibit different levels of conformity. Indeed, generational changes have occurred for Aschian conformity (29) and possibly in frequency-dependent copying of baby names (12). Our stochastic model may therefore produce more realistic population dynamics than previous deterministic models, and comparisons between the two can suggest when the latter is a reasonable approximation to the former.We also allow the number of role models, nt, to vary over time. Agent-based conformity models have incorporated temporal (43) and individual (43, 45, 46) variation in the number of sampled individuals, whereas here, all members of generation t sample the same number nt of role models. Causes of variation in nt are not explored here, but there could be several. For instance, different generations of animals may sample different numbers of role models due to variation in population density. In humans, changes in the use of social media platforms or their features may cause temporal changes in the number of observed individuals. For example, when Facebook added the feature “People You May Know,” the rate of new Facebook connections in a New Orleans dataset nearly doubled (52).In the stochastic model without selection, regardless of the fluctuation in the conformity coefficient(s), if there is conformity on average, the population converges to one of the three equilibria present in Boyd and Richerson’s (4) model with conformity (D(j)>0 for n2<j<n in Eq. 19). These are p*=1 (fixation on variant A), p*=0 (fixation on variant B), and p*=12 (equal representation of A and B). However, their stability properties may differ from those in the deterministic case. In Boyd and Richerson’s (4) model with random copying, every initial frequency p0 is an equilibrium. Here, with random copying expected and independent conformity coefficients, there is convergence to p*=0,12, or 1. In this case, and in the case with conformity expected, convergence to p*=0,12, or 1 also holds with stochastic variation in the number of role models, nt. With either stochastic or constant weak selection in Boyd and Richerson’s (4) simplest model (Eq. 1) and random copying expected, there is convergence to a fixation state (p*=0 or 1). Finally, with anticonformity in the deterministic model or anticonformity expected in the stochastic model, nonconvergence can occur.     

Dynamic rupture initiation and propagation in a fluid-injection laboratory setup with diagnostics across multiple temporal scales

Fluids are known to trigger a broad range of slip events, from slow, creeping transients to dynamic earthquake ruptures. Yet, the detailed mechanics underlying these processes and the conditions leading to different rupture behaviors are not well understood. Here, we use a laboratory earthquake setup, capable of injecting pressurized fluids, to compare the rupture behavior for different rates of fluid injection, slow (megapascals per hour) versus fast (megapascals per second). We find that for the fast injection rates, dynamic ruptures are triggered at lower pressure levels and over spatial scales much smaller than the quasistatic theoretical estimates of nucleation sizes, suggesting that such fast injection rates constitute dynamic loading. In contrast, the relatively slow injection rates result in gradual nucleation processes, with the fluid spreading along the interface and causing stress changes consistent with gradually accelerating slow slip. The resulting dynamic ruptures propagating over wetted interfaces exhibit dynamic stress drops almost twice as large as those over the dry interfaces. These results suggest the need to take into account the rate of the pore-pressure increase when considering nucleation processes and motivate further investigation on how friction properties depend on the presence of fluids.

The close connection between fluids and faulting has been revealed by a large number of observations, both in tectonic settings and during human activities, such as wastewater disposal associated with oil and gas extraction, geothermal energy production, and CO₂ sequestration (1–11). On and around tectonic faults, fluids also naturally exist and are added at depths due to rock-dehydration reactions (12–15) Fluid-induced slip behavior can range from earthquakes to slow, creeping motion. It has long been thought that creeping and seismogenic fault zones have little to no spatial overlap. Nonetheless, growing evidence suggests that the same fault areas can exhibit both slow and dynamic slip (16–19). The existence of large-scale slow slip in potentially seismogenic areas has been revealed by the presence of transient slow-slip events in subduction zones (16, 18) and proposed by studies investigating the physics of foreshocks (20–22).Numerical and laboratory modeling has shown that such complex fault behavior can result from the interaction of fluid-related effects with the rate-and-state frictional properties (9, 14, 19, 23, 24); other proposed rheological explanations for complexities in fault stability include combinations of brittle and viscous rheology (25) and friction-to-flow transitions (26). The interaction of frictional sliding and fluids results in a number of coupled and competing mechanisms. The fault shear resistance τres is typically described by a friction model that linearly relates it to the effective normal stress σ^n via a friction coefficient f:τres=f σ^n=f (σn−p),[1]where σn is the normal stress acting across the fault and p is the pore pressure. Clearly, increasing pore pressure p would reduce the fault frictional resistance, promoting the insurgence of slip. However, such slip need not be fast enough to radiate seismic waves, as would be characteristic of an earthquake, but can be slow and aseismic. In fact, the critical spatial scale h* for the slipping zone to reach in order to initiate an unstable, dynamic event is inversely proportional to the effective normal stress (27, 28) and hence increases with increasing pore pressure, promoting stable slip. This stabilizing effect of increasing fluid pressure holds for both linear slip-weakening and rate-and-state friction; it occurs because lower effective normal stress results in lower fault weakening during slip for the same friction properties. For example, the general form for two-dimensional (2D) theoretical estimates of this so-called nucleation size, h*, on rate-and-state faults with steady-state, velocity-weakening friction is given by:h*=(μ*DRS)/[F(a,b)(σn−p)],[2]where μ*=μ/(1−ν) for modes I and II, and μ*=μ for mode III (29); DRS is the characteristic slip distance; and F(a, b) is a function of the rate-and-state friction parameters a and b. The function F(a, b) depends on the specific assumptions made to obtain the estimate: FRR(a,b)=4(b−a)/π (ref. 27, equation 40) for a linearized stability analysis of steady sliding, or FRA(a,b)=[π(b−a)2]/2b, with a/b>1/2 for quasistatic crack-like expansion of the nucleation zone (ref. 30, equation 42).Hence, an increase in pore pressure induces a reduction in the effective normal stress, which both promotes slip due to lower frictional resistance and increases the critical length scale h*, potentially resulting in slow, stable fault slip instead of fast, dynamic rupture. Indeed, recent field and laboratory observations suggest that fluid injection triggers slow slip first (4, 9, 11, 31). Numerical modeling based on these effects, either by themselves or with an additional stabilizing effect of shear-layer dilatancy and the associated drop in fluid pressure, have been successful in capturing a number of properties of slow-slip events observed on natural faults and in field fluid-injection experiments (14, 24, 32–34). However, understanding the dependence of the fault response on the specifics of pore-pressure increase remains elusive. Several studies suggest that the nucleation size can depend on the loading rate (35–38), which would imply that the nucleation size should also depend on the rate of friction strength change and hence on the rate of change of the pore fluid pressure. The dependence of the nucleation size on evolving pore fluid pressure has also been theoretically investigated (39). However, the commonly used estimates of the nucleation size (Eq. 2) have been developed for faults under spatially and temporally uniform effective stress, which is clearly not the case for fluid-injection scenarios. In addition, the friction properties themselves may change in the presence of fluids (40–42). The interaction between shear and fluid effects can be further affected by fault-gauge dilation/compaction (40, 43–45) and thermal pressurization of pore fluids (42, 46–48).Recent laboratory investigations have been quite instrumental in uncovering the fundamentals of the fluid-faulting interactions (31, 45, 49–57). Several studies have indicated that fluid-pressurization rate, rather than injection volume, controls slip, slip rate, and stress drop (31, 49, 57). Rapid fluid injection may produce pressure heterogeneities, influencing the onset of slip. The degree of heterogeneity depends on the balance between the hydraulic diffusion rate and the fluid-injection rate, with higher injection rates promoting the transition from drained to locally undrained conditions (31). Fluid pressurization can also interact with friction properties and produce dynamic slip along rate-strengthening faults (50, 51).In this study, we investigate the relation between the rate of pressure increase on the fault and spontaneous rupture nucleation due to fluid injection by laboratory experiments in a setup that builds on and significantly develops the previous generations of laboratory earthquake setup of Rosakis and coworkers (58, 59). The previous versions of the setup have been used to study key features of dynamic ruptures, including sub-Rayleigh to supershear transition (60); rupture directionality and limiting speeds due to bimaterial effects (61); pulse-like versus crack-like behavior (62); opening of thrust faults (63); and friction evolution (64). A recent innovation in the diagnostics, featuring ultrahigh-speed photography in conjunction with digital image correlation (DIC) (65), has enabled the quantification of the full-field behavior of dynamic ruptures (66–68), as well as the characterization of the local evolution of dynamic friction (64, 69). In these prior studies, earthquake ruptures were triggered by the local pressure release due to an electrical discharge. This nucleation procedure produced only dynamic ruptures, due to the nearly instantaneous normal stress reduction.To study fault slip triggered by fluid injection, we have developed a laboratory setup featuring a hydraulic circuit capable of injecting pressurized fluid onto the fault plane of a specimen and a set of experimental diagnostics that enables us to detect both slow and fast fault slip and stress changes. The range of fluid-pressure time histories produced by this setup results in both quasistatic and dynamic rupture nucleation; the diagnostics allows us to capture the nucleation processes, as well as the resulting dynamic rupture propagation. In particular, here, we explore two injection techniques: procedure 1, a gradual, and procedure 2, a sharp fluid-pressure ramp-up. An array of strain gauges, placed on the specimen’s surface along the fault, can capture the strain (translated into stress) time histories over a wide range of temporal scales, spanning from microseconds to tens of minutes. Once dynamic ruptures nucleate, an ultrahigh-speed camera records images of the propagating ruptures, which are turned into maps of full-field displacements, velocities, and stresses by a tailored DIC) analysis. One advantage of using a specimen made of an analog material, such as poly(methyl meth-acrylate) (PMMA) used in this study, is its transparency, which allows us to look at the interface through the bulk and observe fluid diffusion over the interface. Another important advantage of using PMMA is that its much lower shear modulus results in much smaller nucleation sizes h* than those for rocks, allowing the experiments to produce both slow and fast slip in samples of manageable sizes.We start by describing the laboratory setup and the diagnostics monitoring the pressure evolution and the slip behavior. We then present and discuss the different slip responses measured as a result of slow versus fast fluid injection and interpret our measurements by using the rate-and-state friction framework and a pressure-diffusion model.     

Thermodynamic controls on rates of iron oxide reduction by extracellular electron shuttles

Anaerobic microbial respiration in suboxic and anoxic environments often involves particulate ferric iron (oxyhydr-)oxides as terminal electron acceptors. To ensure efficient respiration, a widespread strategy among iron-reducing microorganisms is the use of extracellular electron shuttles (EES) that transfer two electrons from the microbial cell to the iron oxide surface. Yet, a fundamental understanding of how EES–oxide redox thermodynamics affect rates of iron oxide reduction remains elusive. Attempts to rationalize these rates for different EES, solution pH, and iron oxides on the basis of the underlying reaction free energy of the two-electron transfer were unsuccessful. Here, we demonstrate that broadly varying reduction rates determined in this work for different iron oxides and EES at varying solution chemistry as well as previously published data can be reconciled when these rates are instead related to the free energy of the less exergonic (or even endergonic) first of the two electron transfers from the fully, two-electron reduced EES to ferric iron oxide. We show how free energy relationships aid in identifying controls on microbial iron oxide reduction by EES, thereby advancing a more fundamental understanding of anaerobic respiration using iron oxides.

The use of iron oxides as terminal electron acceptors in anaerobic microbial respiration is central to biogeochemical element cycling and pollutant transformations in many suboxic and anoxic environments (1–6). To ensure efficient electron transfer to solid-phase ferric iron, Fe(III), at circumneutral pH, metal-reducing microorganisms from diverse phylae use dissolved extracellular electron shuttle (EES), including quinones (7–9), flavins (10–16), and phenazines (17–19), to transfer two electrons per EES molecule from the respiratory chain proteins in the outer membrane of the microbial cell to the iron oxide (17, 20, 21). The oxidized EES can diffuse back to the cell surface for rereduction, thereby completing the catalytic redox cycle involving the EES.The electron transfer from the reduced EES to Fe(III) is considered a key step in overall microbial Fe(III) respiration. Several lines of evidence suggest that the free energy of the electron transfer reaction, ΔrG, controls Fe(III) reduction rates (15, 17, 22, 23). For instance, microbial Fe(III) oxide reduction by dissolved model quinones as EES was accelerated only for quinones with standard two-electron reduction potentials, EH,1,20, that fell into a relatively narrow range of −180±80 mV at pH 7 (24). Furthermore, in abiotic experiments, Fe(III) reduction rates by EES decreased with increasing ΔrG that resulted from increasing either EH,1,20 of the EES (25, 26), the concentration of Fe(II) in the system (27), or solution pH (25, 26, 28). However, substantial efforts to relate Fe(III) reduction rates for different EES species, iron oxides, and pH to the EH,1,20 averaged over both electrons transferred from the EES to the iron oxides were only partially successful (25, 28). Reaction free energies of complex redox processes involving the transfer of multiple electrons can readily be calculated using differences in the reduction potentials averaged over all electrons transferred, and this approach is well established in biogeochemistry and microbial ecology. For kinetic considerations, however, the use of averaged reduction potentials is inappropriate.Herein, we posit that rates of Fe(III) reduction by EES instead relate to the ΔrG of the less exergonic first one-electron transfer from the two-electron reduced EES species to the iron oxide, following the general notion that reaction rates scale with reaction free energies (29). Our hypothesis is based on the fact that, at circumneutral to acidic pH and for many EES, the reduction potential of the first electron transferred to the fully oxidized EES to form the one-electron reduced intermediate semiquinone species, EH,1, is lower than the reduction potential of the second electron transferred to the semiquinone to form the fully two-electron reduced EES species, EH,2 [i.e., EH,1<EH,2 (30–33)]. This difference in one-electron reduction potentials implies that the two-electron reduced EES (i.e., the hydroquinone) is the weaker one-electron reductant for Fe(III) as compared to the semiquinone species. We therefore expect that rates of iron oxide reduction relate to the ΔrG of the first electron transferred from the hydroquinone to Fe(III). The ΔrG of this first electron transfer may even be endergonic provided that the two-electron transfer is exergonic.We verified our hypothesis in abiotic model systems by demonstrating that reduction rates of two geochemically important crystalline iron oxides, goethite and hematite, by two-electron reduced quinone- and flavin-based EES over a wide pH range, and therefore thermodynamic driving force for Fe(III) reduction, correlate with the ΔrG of the first electron transferred from the fully reduced EES to Fe(III). We further show that rates of goethite and hematite reduction by EES reported in the literature are in excellent agreement with our rate data when comparing rates on the basis of the thermodynamics of the less exergonic first of the two electron transfers.     

First-order antiferromagnetic transitions of SrMn2P2 and CaMn2P2 single crystals containing corrugated-honeycomb Mn sublattices

SrMn₂P₂ and CaMn₂P₂ are insulators that adopt the trigonal CaAl₂Si₂-type structure containing corrugated Mn honeycomb layers. Magnetic susceptibility χ and heat capacity versus temperature T data reveal a weak first-order antiferromagnetic (AFM) transition at the Néel temperature TN=53(1) K for SrMn₂P₂ and a strong first-order AFM transition at TN=69.8(3) K for CaMn₂P₂. Both compounds exhibit isotropic and nearly T-independent χ(T≤TN), suggesting magnetic structures in which nearest-neighbor moments are aligned at ≈120° to each other. The ³¹P NMR measurements confirm the strong first-order transition in CaMn₂P₂ but show critical slowing down above TN for SrMn₂P₂, thus also evidencing second-order character. The ³¹P NMR measurements indicate that the AFM structure of CaMn₂P₂ is commensurate with the lattice whereas that of SrMn₂P₂ is incommensurate. These first-order AFM transitions are unique among the class of (Ca, Sr, Ba)Mn₂ (P, As, Sb, Bi)₂ compounds that otherwise exhibit second-order AFM transitions. This result challenges our understanding of the circumstances under which first-order AFM transitions occur.

The Mn-based 122-type pnictides AMn2Pn2 (A= Ca, Sr, Ba; Pn = P, As, Sb, Bi) have received attention owing to their close stoichiometric 122-type relationship to high-Tc iron pnictides. The undoped Mn pnictides are local-moment antiferromagnetic (AFM) insulators like the high-Tc cuprate parent compounds (1–3). The BaMn2Pn2 compounds crystallize in the body-centered tetragonal ThCr2Si2 structure as in AFe2As2 (A = Ca, Sr, Ba, Eu), whereas the (Ca,Sr)Mn2Pn2 compounds crystallize in the trigonal CaAl2Si2-type structure (4). Recently, density-functional theory (DFT) calculations for the 122 pnictide family have suggested that the trigonal 122 transition-metal pnictides that have the CaAl2Si2 structure might compose a new family of magnetically frustrated materials in which to study the potential superconducting mechanism (5, 6). It had previously been suggested on theoretical grounds that CaMn₂Sb₂ is a fully frustrated classical magnetic system arising from proximity to a tricritical point (7–9).The electrical resistivity ρ and heat capacity Cp versus temperature T of single-crystal CaMn₂P₂ were reported in ref. 10. The compound is an insulator at T = 0 and undergoes a first-order transition of some type at 69.5 K. The Raman spectrum of CaMn₂P₂ at T = 10 K showed new peaks compared to the spectrum at 300 K, whereas the authors’ single-crystal X-ray diffraction measurements showed no difference in the crystal structure at 293 and 40 K. They suggested that the results of the two types of measurements could be reconciled if a superstructure formed below 69.5 K (10). The authors’ magnetic susceptibility χ(T) measurements below 400 K revealed no evidence for a magnetic transition.Here we report the detailed properties of trigonal CaMn₂P₂ and SrMn₂P₂ (11) single crystals. We present the results of single-crystal X-ray diffraction (XRD), electrical resistivity ρ in the ab plane (hexagonal unit cell) versus temperature T, isothermal magnetization versus applied magnetic field M(H), magnetic susceptibility χ(T), heat capacity Cp(H,T), and ³¹P NMR measurements. We find from Cp(T), χ(T), and NMR that CaMn₂P₂ exhibits a strong first-order AFM transition at TN=69.8(3) K whereas SrMn₂P₂ shows a weak first-order transition at TN=53(1) K but with critical slowing down on approaching TN from above as revealed from NMR, a characteristic feature of second-order transitions. Thus, remarkably, the AFM transition in SrMn₂P₂ has characteristics of both first- and second-order transitions. The χ(T) data also reveal the presence of strong isotropic AFM spin fluctuations in the paramagnetic (PM) state above TN up to our maximum measurement temperatures of 900 and 350 K for SrMn₂P₂ and CaMn₂P₂, respectively. This behavior likely arises from spin fluctuations associated with the quasi–two-dimensional nature of the Mn spin layers (12) together with possible contributions from magnetic frustration. Our single-crystal XRD data at room temperature and high-resolution synchrotron XRD data at T = 20 K for SrMn₂P₂ and CaMn₂P₂ demonstrate conclusively that there is no structure change of either compound on cooling below their respective TN.Our studies of SrMn₂P₂ and CaMn₂P₂ thus identify the only known members of the class of materials with general formula AMn2Pn2 containing Mn2+ spins S = 5/2 that exhibit first-order AFM transitions, where A = Ca, Sr, or Ba and the pnictogen Pn= P, As, Sb, or Bi. In particular, only second-order AFM transitions are found in CaMn2As2 (13), SrMn2As2 (13–15), CaMn2Sb2 (8, 9, 16–19), SrMn2Sb2 (16, 19), and CaMn2Bi2 (20).     

Layering transitions and metastable structures of cholesteric liquid crystals in cylindrical confinement

Our study of cholesteric lyotropic chromonic liquid crystals in cylindrical confinement reveals the topological aspects of cholesteric liquid crystals. The double-twist configurations we observe exhibit discontinuous layering transitions, domain formation, metastability, and chiral point defects as the concentration of chiral dopant is varied. We demonstrate that these distinct layer states can be distinguished by chiral topological invariants. We show that changes in the layer structure give rise to a chiral soliton similar to a toron, comprising a metastable pair of chiral point defects. Through the applicability of the invariants we describe to general systems, our work has broad relevance to the study of chiral materials.

Chiral liquid crystals (LCs) are ubiquitous, useful, and rich systems (1–4). From the first discovery of the liquid crystalline phase to the variety of chiral structures formed by biomolecules (5–9), the twisted structure, breaking both mirror and continuous spatial symmetries, is omnipresent. The unique structure also makes the chiral nematic (cholesteric) LC, an essential material for applications utilizing the tunable, responsive, and periodic modulation of anisotropic properties.The cholesteric is also a popular model system to study the geometry and topology of partially ordered matter. The twisted ground state of the cholesteric is often incompatible with confinement and external fields, exhibiting a large variety of frustrated and metastable director configurations accompanying topological defects. Besides the classic example of cholesterics in a Grandjean−Cano wedge (10, 11), examples include cholesteric droplets (12–16), colloids (17–19), shells (20–22), tori (23, 24), cylinders (25–29), microfabricated structures (30, 31), and films between parallel plates with external fields (32–40). These structures are typically understood using a combination of nematic (achiral) topology (41, 42) and energetic arguments, for example, the highly successful Landau−de Gennes approach (43). However, traditional extensions of the nematic topological approach to cholesterics are known to be conceptually incomplete and difficult to apply in regimes where the system size is comparable to the cholesteric pitch (41, 44).An alternative perspective, chiral topology, can give a deeper understanding of these structures (45–47). In this approach, the key role is played by the twist density, given in terms of the director field n by n⋅∇×n. This choice is not arbitrary; the Frank free energy prefers n⋅∇×n≈−q0=2π/p0 with a helical pitch p0, and, from a geometric perspective, n⋅∇×n≠0 defines a contact structure (48). This allows a number of new integer-valued invariants of chiral textures to be defined (45). A configuration with a single sign of twist is chiral, and two configurations which cannot be connected by a path of chiral configurations are chirally distinct, and hence separated by a chiral energy barrier. Within each chiral class of configuration, additional topological invariants may be defined using methods of contact topology (45–48), such as layer numbers. Changing these chiral topological invariants requires passing through a nonchiral configuration. Cholesterics serve as model systems for the exploration of chirality in ordered media, and the phenomena we describe here—metastability in chiral systems controlled by chiral topological invariants—has applicability to chiral order generally. This, in particular, includes chiral ferromagnets, where, for example, our results on chiral topological invariants apply to highly twisted nontopological Skyrmions (49, 50) (“Skyrmionium”).Our experimental model to explore the chiral topological invariants is the cholesteric phase of lyotropic chromonic LCs (LCLCs). The majority of experimental systems hitherto studied are based on thermotropic LCs with typical elastic and surface-anchoring properties. The aqueous LCLCs exhibiting unusual elastic properties, that is, very small twist modulus K2 and large saddle-splay modulus K24 (51–56), often leading to chiral symmetry breaking of confined achiral LCLCs (53, 54, 56–61), may enable us to access uncharted configurations and defects of topological interests. For instance, in the layer configuration by cholesteric LCLCs doped with chiral molecules, their small K2 provides energetic flexibility to the thickness of the cholesteric layer, that is, the repeating structure where the director n twists by π. The large K24 affords curvature-induced surface interactions in combination with a weak anchoring strength of the lyotropic LCs (62–64).We present a systematic investigation of the director configuration of cholesteric LCLCs confined in cylinders with degenerate planar anchoring, depending on the chiral dopant concentration. We show that the structure of cholesteric configurations is controlled by higher-order chiral topological invariants. We focus on two intriguing phenomena observed in cylindrically confined cholesterics. First, the cylindrical symmetry renders multiple local minima to the energy landscape and induces discontinuous increase of twist angles, that is, a layering transition, upon the dopant concentration increase. Additionally, the director configurations of local minima coexist as metastable domains with point-like defects between them. We demonstrate that a chiral layer number invariant distinguishes these configurations, protects the distinct layer configurations (45), and explains the existence of the topological defect where the invariant changes.     

Structures and topological defects in pressure-driven lyotropic chromonic liquid crystals

Lyotropic chromonic liquid crystals are water-based materials composed of self-assembled cylindrical aggregates. Their behavior under flow is poorly understood, and quantitatively resolving the optical retardance of the flowing liquid crystal has so far been limited by the imaging speed of current polarization-resolved imaging techniques. Here, we employ a single-shot quantitative polarization imaging method, termed polarized shearing interference microscopy, to quantify the spatial distribution and the dynamics of the structures emerging in nematic disodium cromoglycate solutions in a microfluidic channel. We show that pure-twist disclination loops nucleate in the bulk flow over a range of shear rates. These loops are elongated in the flow direction and exhibit a constant aspect ratio that is governed by the nonnegligible splay-bend anisotropy at the loop boundary. The size of the loops is set by the balance between nucleation forces and annihilation forces acting on the disclination. The fluctuations of the pure-twist disclination loops reflect the tumbling character of nematic disodium cromoglycate. Our study, including experiment, simulation, and scaling analysis, provides a comprehensive understanding of the structure and dynamics of pressure-driven lyotropic chromonic liquid crystals and might open new routes for using these materials to control assembly and flow of biological systems or particles in microfluidic devices.

Lyotropic chromonic liquid crystals (LCLCs) are aqueous dispersions of organic disk-like molecules that self-assemble into cylindrical aggregates, which form nematic or columnar liquid crystal phases under appropriate conditions of concentration and temperature (1–6). These materials have gained increasing attention in both fundamental and applied research over the past decade, due to their distinct structural properties and biocompatibility (4, 7–14). Used as a replacement for isotropic fluids in microfluidic devices, nematic LCLCs have been employed to control the behavior of bacteria and colloids (13, 15–20).Nematic liquid crystals form topological defects under flow, which gives rise to complex dynamical structures that have been extensively studied in thermotropic liquid crystals (TLCs) and liquid crystal polymers (LCPs) (21–29). In contrast to lyotropic liquid crystals that are dispersed in a solvent and whose phase can be tuned by either concentration or temperature, TLCs do not need a solvent to possess a liquid-crystalline state and their phase depends only on temperature (30). Most TLCs are shear-aligned nematics, in which the director evolves toward an equilibrium out-of-plane polar angle. Defects nucleate beyond a critical Ericksen number due to the irreconcilable alignment of the directors from surface anchoring and shear alignment in the bulk flow (24, 31–33). With an increase in shear rate, the defect type can transition from π-walls (domain walls that separate regions whose director orientation differs by an angle of π) to ordered disclinations and to a disordered chaotic regime (34). Recent efforts have aimed to tune and control the defect structures by understanding the relation between the selection of topological defect types and the flow field in flowing TLCs. Strategies to do so include tuning the geometry of microfluidic channels, inducing defect nucleation through the introduction of isotropic phases or designing inhomogeneities in the surface anchoring (35–39). LCPs are typically tumbling nematics for which α2α3 < 0, where α2 and α3 are the Leslie viscosities. This leads to a nonzero viscous torque for any orientation of the director, which allows the director to rotate in the shear plane (22, 29, 30, 40). The tumbling character of LCPs facilitates the nucleation of singular topological defects (22, 40). Moreover, the molecular rotational relaxation times of LCPs are longer than those of TLCs, and they can exceed the timescales imposed by the shear rate. As a result, the rheological behavior of LCPs is governed not only by spatial gradients of the director field from the Frank elasticity, but also by changes in the molecular order parameter (25, 41–43). With increasing shear rate, topological defects in LCPs have been shown to transition from disclinations to rolling cells and to worm-like patterns (25, 26, 43).Topological defects occurring in the flow of nematic LCLCs have so far received much more limited attention (44, 45). At rest, LCLCs exhibit unique properties distinct from those of TLCs and LCPs (1, 2, 4–6, 44). In particular, LCLCs have significant elastic anisotropy compared to TLCs; the twist Frank elastic constant, K2, is much smaller than the splay and bend Frank elastic constants, K1 and K3. The resulting relative ease with which twist deformations can occur can lead to a spontaneous symmetry breaking and the emergence of chiral structures in static LCLCs under spatial confinement, despite the achiral nature of the molecules (4, 46–51). When driven out of equilibrium by an imposed flow, the average director field of LCLCs has been reported to align predominantly along the shear direction under strong shear but to reorient to an alignment perpendicular to the shear direction below a critical shear rate (52–54). A recent study has revealed a variety of complex textures that emerge in simple shear flow in the nematic LCLC disodium cromoglycate (DSCG) (44). The tumbling nature of this liquid crystal leads to enhanced sensitivity to shear rate. At shear rates γ˙<1 s−1, the director realigns perpendicular to the flow direction adapting a so-called log-rolling state characteristic of tumbling nematics. For 1 s−1<γ˙<10 s−1, polydomain textures form due to the nucleation of pure-twist disclination loops, for which the rotation vector is parallel to the loop normal, and mixed wedge-twist disclination loops, for which the rotation vector is perpendicular to the loop normal (44, 55). Above γ˙>10 s−1, the disclination loops gradually transform into periodic stripes in which the director aligns predominantly along the flow direction (44).Here, we report on the structure and dynamics of topological defects occurring in the pressure-driven flow of nematic DSCG. A quantitative evaluation of such dynamics has so far remained challenging, in particular for fast flow velocities, due to the slow image acquisition rate of current quantitative polarization-resolved imaging techniques. Quantitative polarization imaging traditionally relies on three commonly used techniques: fluorescence confocal polarization microscopy, polarizing optical microscopy, and LC-Polscope imaging. Fluorescence confocal polarization microscopy can provide accurate maps of birefringence and orientation angle, but the fluorescent labeling may perturb the flow properties (56). Polarizing optical microscopy requires a mechanical rotation of the polarizers and multiple measurements, which severely limits the imaging speed. LC-Polscope, an extension of conventional polarization optical microscopy, utilizes liquid crystal universal compensators to replace the compensator used in conventional polarization microscopes (57). This leads to an enhanced imaging speed and better compensation for polarization artifacts of the optical system. The need for multiple measurements to quantify retardance, however, still limits the acquisition rate of LC-Polscopes.We overcome these challenges by using a single-shot quantitative polarization microscopy technique, termed polarized shearing interference microscopy (PSIM). PSIM combines circular polarization light excitation with off-axis shearing interferometry detection. Using a custom polarization retrieval algorithm, we achieve single-shot mapping of the retardance, which allows us to reach imaging speeds that are limited only by the camera frame rate while preserving a large field-of-view and micrometer spatial resolution. We provide a brief discussion of the optical design of PSIM in Materials and Methods; further details of the measurement accuracy and imaging performance of PSIM are reported in ref. 58.Using a combination of experiments, numerical simulations and scaling analysis, we show that in the pressure-driven flow of nematic DSCG solutions in a microfluidic channel, pure-twist disclination loops emerge for a certain range of shear rates. These loops are elongated in the flow with a fixed aspect ratio. We demonstrate that the disclination loops nucleate at the boundary between regions where the director aligns predominantly along the flow direction close to the channel walls and regions where the director aligns predominantly perpendicular to the flow direction in the center of the channel. The large elastic stresses of the director gradient at the boundary are then released by the formation of disclination loops. We show that both the characteristic size and the fluctuations of the pure-twist disclination loops can be tuned by controlling the flow rate.     

Rapidly descending dark energy and the end of cosmic expansion

If dark energy is a form of quintessence driven by a scalar field ϕ evolving down a monotonically decreasing potential V(ϕ) that passes sufficiently below zero, the universe is destined to undergo a series of smooth transitions. The currently observed accelerated expansion will cease; soon thereafter, expansion will come to end altogether; and the universe will pass into a phase of slow contraction. In this paper, we consider how short the remaining period of expansion can be given current observational constraints on dark energy. We also discuss how this scenario fits naturally with cyclic cosmologies and recent conjectures about quantum gravity.

In the Λ cold dark matter (Λ CDM) model, dark energy takes the form of a positive cosmological constant, in which case the current period of accelerated expansion will endure indefinitely into the future (1). An alternative is that the current vacuum is metastable and has positive energy density. If it is separated by an energy barrier from a true vacuum phase with zero or negative vacuum density, then accelerated expansion will be ended by the nucleation of a bubble of true vacuum that grows to encompass us. Until that moment, cosmological observations will be indistinguishable from the Λ CDM picture. Without extreme fine tuning, the timescale before a bubble will nucleate (2) and pass our location can be exponentially many Hubble times in the future (for example, refs. 3 and 4). (Here and throughout, “Hubble time” refers to H0−1≈14 Gy, where H₀ is the current Hubble expansion rate.) Also, the ultrarelativistic bubble wall will likely destroy all observers in its path, so there will be no surviving witnesses to the end of accelerated expansion (2).A third possibility, to be considered here, is that the dark energy is a type of quintessence due to a scalar field ϕ evolving down a monotonically decreasing potential V(ϕ) (5). Since the current value of V(ϕ0) is extraordinarily small today as measured in Planck mass units, there is a wide range of forms for V(ϕ) that pass through zero and continue to large negative values where V(ϕ)≪−V(ϕ0). In this case, the equations of motion of Einstein’s general theory of relativity dictate that the universe is destined to undergo a remarkable series of smooth transitions (6–8).First, as the positive potential energy density decreases and the kinetic energy density comes to exceed it, the current phase of accelerated expansion will end and smoothly transition to a period of decelerated expansion. Next, as the scalar field continues to evolve down the potential, the potential energy density will become sufficiently negative that the total energy density [∝H2(t)] and, consequently, the Hubble parameter H(t), will reach zero. Consequently, expansion (H > 0) will stop altogether and smoothly change to contraction (H < 0). More precisely, the transition will be to a phase of slow contraction (7, 8) in which the Friedmann–Robertson–Walker (FRW) scale factor a(t)∝|H−1|α, where α<1/3.In this paper, we consider how soon these transitions could begin. That is, What is the minimal time, beginning from the present (t = t₀), before expansion ends and contraction begins given current observational constraints on dark energy and without introducing extreme fine tuning? One might imagine the answer is one or more Hubble times given how well Λ CDM is claimed to fit current cosmological data.     

Unusual magnetotransport in twisted bilayer graphene

We present transport measurements of bilayer graphene with a 1.38^∘ interlayer twist. As with other devices with twist angles substantially larger than the magic angle of 1.1^∘, we do not observe correlated insulating states or band reorganization. However, we do observe several highly unusual behaviors in magnetotransport. For a large range of densities around half filling of the moiré bands, magnetoresistance is large and quadratic. Over these same densities, the magnetoresistance minima corresponding to gaps between Landau levels split and bend as a function of density and field. We reproduce the same splitting and bending behavior in a simple tight-binding model of Hofstadter’s butterfly on a triangular lattice with anisotropic hopping terms. These features appear to be a generic class of experimental manifestations of Hofstadter’s butterfly and may provide insight into the emergent states of twisted bilayer graphene.

The mesmerizing Hofstadter butterfly spectrum arises when electrons in a two-dimensional periodic potential are immersed in an out-of-plane magnetic field. When the magnetic flux Φ through a unit cell is a rational multiple p / q of the magnetic flux quantum Φ0=h/e, each Bloch band splits into q subbands (1). The carrier densities corresponding to gaps between these subbands follow straight lines when plotted as a function of normalized density n/ns and magnetic field (2). Here, n_s is the density of carriers required to fill the (possibly degenerate) Bloch band. These lines can be described by the Diophantine equation (n/ns)=t(Φ/Φ0)+s for integers s and t. In experiments, they appear as minima or zeros in longitudinal resistivity coinciding with Hall conductivity quantized at σxy=te2/h (3, 4). Hofstadter originally studied magnetosubbands emerging from a single Bloch band on a square lattice. In the following decades, other authors considered different lattices (5–7), the effect of anisotropy (6, 8–10), next-nearest-neighbor hopping (11–15), interactions (16, 17), density wave states (9), and graphene moirés (18, 19).It took considerable ingenuity to realize clean systems with unit cells large enough to allow conventional superconducting magnets to reach Φ/Φ0∼1. The first successful observation of the butterfly in electrical transport measurements was in GaAs/AlGaAs heterostructures with lithographically defined periodic potentials (20–22). These experiments demonstrated the expected quantized Hall conductance in a few of the largest magnetosubband gaps. In 2013, three groups mapped out the full butterfly spectrum in both density and field in heterostructures based on monolayer (23, 24) and bilayer (25) graphene. In all three cases, the authors made use of the 2% lattice mismatch between their graphene and its encapsulating hexagonal boron nitride (hBN) dielectric. With these layers rotationally aligned, the resulting moiré pattern was large enough in area that gated structures studied in available high-field magnets could simultaneously approach normalized carrier densities and magnetic flux ratios of 1. Later work on hBN-aligned bilayer graphene showed that, likely because of electron–electron interactions, the gaps could also follow lines described by fractional s and t (26).In twisted bilayer graphene (TBG), a slight interlayer rotation creates a similar-scale moiré pattern. Unlike with graphene–hBN moirés, in TBG there is a gap between lowest and neighboring moiré subbands (27). As the twist angle approaches the magic angle of 1.1^∘ the isolated moiré bands become flat (28, 29), and strong correlations lead to fascinating insulating (30–37), superconducting (31–33, 35–37), and magnetic (34, 35, 38) states. The strong correlations tend to cause moiré subbands within a fourfold degenerate manifold to move relative to each other as one tunes the density, leading to Landau levels that project only toward higher magnitude of density from charge neutrality and integer filling factors (37, 39). This correlated behavior obscures the single-particle Hofstadter physics that would otherwise be present.In this work, we present measurements from a TBG device twisted to 1.38^∘. When we apply a perpendicular magnetic field, a complicated and beautiful fan diagram emerges. In a broad range of densities on either side of charge neutrality, the device displays large, quadratic magnetoresistance. Within the magnetoresistance regions, each Landau level associated with ν=±8,±12,±16,… appears to split into a pair, and these pairs follow complicated paths in field and density, very different from those predicted by the usual Diophantine equation. Phenomenology similar in all qualitative respects appears in measurements on several regions of this same device with similar twist angles and in two separate devices, one at 1.59^∘ and the other at 1.70^∘ (see SI Appendix for details).We reproduce the unusual features of the Landau levels (LLs) in a simple tight-binding model on a triangular lattice with anisotropy and a small energetic splitting between two species of fermions. At first glance, this is surprising, because that model does not represent the symmetries of the experimental moiré structure. We speculate that the unusual LL features we experimentally observe can generically emerge from spectra of Hofstadter models that include the same ingredients we added to the triangular lattice model. With further theoretical work it may be possible to use our measurements to gain insight into the underlying Hamiltonian of TBG near the magic angle.     

Lithium superoxide encapsulated in a benzoquinone anion matrix

Lithium peroxide is the crucial storage material in lithium–air batteries. Understanding the redox properties of this salt is paramount toward improving the performance of this class of batteries. Lithium peroxide, upon exposure to p–benzoquinone (p–C₆H₄O₂) vapor, develops a deep blue color. This blue powder can be formally described as [Li₂O₂]0.3 · [LiO₂]0.7 · {Li[p–C₆H₄O₂]}0.7, though spectroscopic characterization indicates a more nuanced structural speciation. Infrared, Raman, electron paramagnetic resonance, diffuse-reflectance ultraviolet-visible and X-ray absorption spectroscopy reveal that the lithium salt of the benzoquinone radical anion forms on the surface of the lithium peroxide, indicating the occurrence of electron and lithium ion transfer in the solid state. As a result, obligate lithium superoxide is formed and encapsulated in a shell of Li[p–C₆H₄O₂] with a core of Li₂O₂. Lithium superoxide has been proposed as a critical intermediate in the charge/discharge cycle of Li–air batteries, but has yet to be isolated, owing to instability. The results reported herein provide a snapshot of lithium peroxide/superoxide chemistry in the solid state with redox mediation.

The advent of metal–air batteries has provided an impetus for understanding the structure, spectroscopic properties, and chemical reactivity of various metal oxides. Lithium–air batteries, which possess a theoretical energy density approaching that of liquid fuels, have emerged as potential candidates to replace lithium-ion batteries (1–4). Lithium–air batteries operate by electron transfer from a high-surface-area cathode to oxygen gas during discharge, generating lithium peroxide deposits. Upon charging, the lithium peroxide is oxidized back to oxygen gas. Despite demonstrating promise as a replacement for lithium-ion batteries, this electrochemical energy-storage system suffers from numerous challenges that must be overcome (5–8), the most important of which is reversible charging.Lithium superoxide, LiO₂, is an important intermediate in both the reduction of oxygen to lithium peroxide and oxidation of lithium peroxide back to oxygen (9–13). Lithium superoxide, via disproportionation, is thought to be responsible for the growth of large lithium peroxide toroids commonly observed during discharge of nonaqueous lithium–air cells, but is also implicated in numerous studies (14–18) as being responsible, either directly or through the intermediacy of ¹O₂ (19, 20), for the degradation of the organic solvent and electrolyte in the battery. Furthermore, “superoxide-like” sites on the surface of lithium peroxide are thought to be responsible for both enhanced reactivity with electrolytes in lithium–air batteries and enhanced conductivity (21, 22).While the superoxide salts of cesium, rubidium, potassium, and sodium are well known, and potassium superoxide is commercially available, the lithium salt of superoxide was not observed definitively until recently in noble gas matrices at low temperatures (23–27). Matrix isolation studies have demonstrated that the infrared (IR) absorption peak associated with the O–O stretch of superoxide loses ∼80% of its intensity when warmed from 15 to 34 K. Two additional reports of the cryogenic preparation of LiO₂ exist: one with O₃ and Li₂O₂ in freon at −65 °C (28) and another involving treatment of O₂ with lithium metal in NH₃ at −78 °C (29).Strides have also been made toward observing or directly stabilizing lithium superoxide in lithium–air batteries (30–32). At room temperature, lithium superoxide, if unstabilized, is expected to have a fleeting existence, prompting a study to experimentally demonstrate that Raman signals assigned to lithium superoxide in earlier lithium–air works may be ascribed to polyvinylidene fluoride (PVDF) binder that has undergone a dehydrohalogenation reaction (33). However, a subsequent report has asserted that rigorous drying of the PVDF binder precludes the dehydrohalogenation reaction (34). Indeed, the presence and involvement of observable lithium superoxide in lithium–air batteries remains a topic of interest and some controversy (35).The continued advancement of lithium–air batteries will require improved characterization and understanding of the properties of lithium superoxide, particularly with respect to elucidating the fundamental properties of the salt in the solid state. Reported herein is a solid-state material that models the oxidation of lithium peroxide to lithium superoxide, with a surface coating of p–benzoquinone acting as the electron acceptor. The system recapitulates the proposal of superoxide-like sites on the surface of lithium peroxide in lithium–air batteries.