Wave vector dependence of the dynamics in supercooled metallic liquids
B. Ruta, S. Hechler, N. Neuber, D. Orsi, L. Cristofolini, O. Gross, B. Bochtler, M. Frey, A. Kuball, S. S. Riegler, M. Stolpe, Z. Evenson, C. Gutt, F. Westermei… 展开作者
B. Ruta, S. Hechler, N. Neuber, D. Orsi, L. Cristofolini, O. Gross, B. Bochtler, M. Frey, A. Kuball, S. S. Riegler, M. Stolpe, Z. Evenson, C. Gutt, F. Westermeier, R. Busch, I. Gallino
We present a detailed investigation of the wave vector dependence of collective atomic motion in Au49Cu26.9Si16.3Ag5.5Pd2.3 and Pd42.5Cu27Ni9.5P21 supercooled liquids close to the glass transition temperature. Using x-ray photon correlation spectroscopy in a precedent uncovered spatial range of only few interatomic distances, we show that the microscopic structural relaxation process follows in phase the structure with a marked slowing down at the main average inter-particle distance. This behavior is accompanied by dramatic changes in the shape of the intermediate scattering functions which suggest the presence of large dynamical heterogeneities at length-scales corresponding to few particle diameters. A ballistic-like mechanism of particle motion seems to govern the structural relaxation of the two systems in the highly viscous phase, likely associated to hopping of caged particles in agreement with theoretical studies.
The mechanism of acoustic wave propagation in supercooled liquids is not yet fully understood since the vibrational dynamics of supercooled liquids are strongly affected by their amorphous inherent structures. In this paper, the acoustic wave propagation in a supercooled model liquid is studied by using normal mode analysis. Due to the highly disordered inherent structure, a single acoustic wave is decomposed into many normal modes in broad frequency range. This causes the rapid decay of the acoustic wave and results in anomalous wavenumber dependency of the dispersion relation and the rate of…
We propose a dynamic structure of coupled dynamic molecular strings for supercooled small polar molecule liquids and accordingly we obtain the Hamiltonian of the rotational degrees of freedom of the system. From the Hamiltonian, the strongly correlated supercooled polar liquid state is renormalized to a normal superdipole (SD) liquid state. This scenario describes the following main features of the primary or a-relaxation dynamics in supercooled polar liquids: (1) the average relaxation time evolves from a high temperature Arrhenius to a low temperature non-Arrhenius or super-Arrhenius behavio…
For a deeply supercooled liquid just above its glass transition temperature, we present a simple thermodynamic model, where the deeply supercooled liquid is assumed to be a mixture of solid-like and liquid-like micro regions. The mole fraction of the liquid-like regions controls the thermodynamic properties of the supercooled liquid while that of the solid-like regions controls its relaxation time or viscosity. From the universal temperature dependence of the molar excess entropy for the deeply supercooled liquids, we derive the temperature dependence of the mole fraction of the liquid-like re…
The cage effect is widely accepted as the basic microscopic mechanism underlying the physics of supercooled liquids in contrast with usual liquids which are governed by molecular interactions only. In this work we implement a new toy model coined to reproduce the cage effect with variants including structural fluctuations and kinetic constraints. We use this new model to investigate which glass-transition features are directly due to the cage effect and which are due to more complex mechanisms.
We study the influence of the system size on various static and dynamic properties of a supercooled binary Lennard-Jones liquid via computer simulations. In this way, we demonstrate that the treatment of systems as small as N=65 particles yields relevant results for the understanding of bulk properties. Especially, we find that a system of N=130 particles behaves basically as two non-interacting systems of half the size.
Usually, supercooled liquids and glasses are thermodynamically unstable against crystallization. Classical nucleation theory (CNT) has been used to describe the crystallization dynamics of supercooled liquids. However, recent studies on overcompressed hard spheres show that their crystallization dynamics are intermittent and mediated by avalanche-like rearrangements of particles, which largely differ from the CNT. These observations suggest that the crystallization times of deeply supercooled liquids or glasses cannot be described by the CNT, but this point has not yet been studied in detail. …
Advances in high-precision dielectric spectroscopy has enabled access to non-linear susceptibilities of polar molecular liquids. The observed non-monotonic behavior has been claimed to provide strong support for theories of dynamic arrest based on thermodynamic amorphous order. Here we approach this question from the perspective of dynamic facilitation, an alternative view focusing on emergent kinetic constraints underlying the dynamic arrest of a liquid approaching its glass transition. We derive explicit expressions for the frequency-dependent higher-order dielectric susceptibilities exhibit…
We report extensive simulations of liquid supercooled states for a simple three-sites molecular model, introduced by Lewis and Wahnstr"om [L. J. Lewis and G. Wahnstr"om, Phys. Rev. E 50, 3865 (1994)] to mimic the behavior of ortho-terphenyl. The large system size and the long simulation length allow to calculate very precisely --- in a large q-vector range --- self and collective correlation functions, providing a clean and simple reference model for theoretical descriptions of molecular liquids in supercooled states. The time and wavevector dependence of the site-site correlation functions ar…
Large scale molecular dynamics simulations are used to investigate the structural and dynamical modifications of supercooled water when confined inside an hydrophilic nanopore. We then investigate the evolution of the auto-organization of the most and the least mobile molecules (dynamical heterogeneity and string-like cooperative motions) when supercooled water is confined. Our calculations use the recent TIP5P intermolecular potential for water. We observe a strong slowing down of the dynamical properties when the liquid is confined, although the liquid structure is found to remain unchanged …
We report a thorough characterization of the glassy phases of mixtures of succinonitrile and glutaronitrile via dielectric spectroscopy and differential scanning calorimetry. This system is revealed to be one of the rare examples where both glassy states of matter, a structurally disordered supercooled liquid and an orientationally disordered plastic crystal, can be prepared in the same material. Both disordered states can be easily supercooled, finally arriving at a structural-glass or a glassy-crystal state. Detailed investigations using broadband dielectric spectroscopy enable a comparison …
We propose deformations of inherent structures as a suitable tool for detecting structural changes underlying the onset of cooperativity in supercooled liquids. The non-affine displacement (NAD) field resulting from the applied deformation shows characteristic differences between the high temperature liquid and supercooled state, that are typically observed in dynamic quantities. The average magnitude of the NAD is very sensitive to temperature changes in the supercooled regime and is found to be strongly correlated with the inherent structure energy. In addition, the NAD field is characterize…
Dielectric relaxation spectroscopy (1 Hz - 20 GHz) has been performed on supercooled glass-formers from the temperature of glass transition (T_g) up to that of melting. Precise measurements particularly in the frequencies of MHz-order have revealed that the temperature dependences of secondary beta-relaxation times deviate from the Arrhenius relation in well above T_g. Consequently, our results indicate that the beta-process merges into the primary alpha-mode around the melting temperature, and not at the dynamical transition point T which is approximately equal to 1.2 T_g.
Slow dynamics of difluorotetrachloroethane in both supercooled plastic crystal and supercooled liquid states have been investigated from Molecular Dynamics simulations. The temperature and wave-vector dependence of collective dynamics in both states are probed using coherent dynamical scattering functions $S(Q,t)$. Our results confirm the strong analogy between molecular liquids and plastic crystals for which $α$-relaxation times and non-ergodicity parameters are controlled by the non trivial static correlations $S(Q)$ as predicted by the Mode Coupling Theory. The use of infinitely thin needle…
We introduce a supercooled liquid model and obtain parameter-free quantitative predictions that are in excellent agreement with numerical simulations, notably in the hard low-temperature region characterized by strong deviations from Mode-Coupling-Theory behavior. The model is the Fredrickson-Andersen Kinetically-Constrained-Model on the three-dimensional $M$-layer lattice. The agreement has implications beyond the specific model considered because the theory is potentially valid for many more systems, including realistic models and actual supercooled liquids.
According to the Random First Order Transition (RFOT) theory of glasses, the barriers for activated dynamics in supercooled liquids vanish as the temperature of a viscous liquid approaches the dynamical transition temperature from below. This occurs due to a decrease of the surface tension between local meta-stable molecular arrangements much like at a spinodal. The dynamical transition thus represents a crossover from the low $T$ activated bevavior to a collisional transport regime at high $T$. This barrier softening explains the deviation of the relaxation times, as a function of temperature…
We have developed a scaled parametric equation of state to describe and predict thermodynamic properties of supercooled water. The equation of state, built on the growing evidence that the critical point of supercooled liquid-liquid water separation exists, is universal in terms of theoretical scaling fields and is shown to belong to the Ising-model class of universality. The theoretical scaling fields are postulated to be analytical combinations of the physical fields, pressure and temperature. The equation of state enables us to accurately locate the "Widom line" (the locus of stability mini…
Large scale Molecular Dynamics simulations of sixty-five systems with N=80000 Lennard-Jones particles at two different supercooled liquid state points reveals, that the supercooled states contain spatially heterogeneous distributed subdomains of particles with significant higher bond-order than the mean bond-order in the supercooled liquid. The onset of crystallization starts in such an area with relative high six-ford bond-order for a supercooled state, but low bond-order for a fcc crystal. The critical nucleus of N \approx 70 particles is surrounded by many hundred of particles with relative…
In recent years it has become widely accepted that a dynamical length scale ξ_α plays an important role in supercooled liquids near the glass transition. We examine the implications of the interplay between the growing ξ_α and the size of the crystal nucleus, ξ_M, which shrinks on cooling. We argue that at low temperatures where ξ_α > ξ_M a new crystallization mechanism emerges enabling rapid development of a large scale web of sparsely connected crystallinity. Though we predict this web percolates the system at too low a temperature to be easily seen in the laboratory, there are noticeable re…
We study a simple and versatile diatomic potential function coined to prevent crystallization in supercooled liquids. We show that the corresponding liquid does not crystallize even with very long simulation runs at the lowest temperature that we can access with ergodic simulations. The medium displays the usual features of supercooled materials and a non-Arrhenius dependence of the diffusion coefficient and alpha relaxation time with temperature. We also observe the breakdown of the Stokes-Einstein relation at low temperatures.
The use of the isoconfigurational ensemble to explore structure-dynamic correlations in supercooled liquids is examined. The statistical error of the dynamic propensity and its spatial distribution are determined. The authors present the spatial distribution of the particle non-Gaussian parameter as a measure of the intermittency with which particles exhibit their propensity for motion. The ensemble average of the direction of particle motion is introduced to establish the anisotropy of the dynamic propensity.
Using computer simulations, we establish that the structure of a supercooled binary atomic liquid mixture consists of common neighbour structures similar to those found in the equilibrium crystal phase, a Laves structure. Despite the large accumulation of crystal-like structure, we establish that the supercooled liquid represents a true metastable liquid and that liquid can borrow crystal structure without being destabilized. We consider whether this feature might be the origin of all instances of liquids of a strongly favoured local structure.
The thermodynamics and dynamics of supercooled liquids correlate with their elasticity. In particular for covalent networks, the jump of specific heat is small and the liquid is {\it strong} near the threshold valence where the network acquires rigidity. By contrast, the jump of specific heat and the fragility are large away from this threshold valence. In a previous work [Proc. Natl. Acad. Sci. U.S.A., 110, 6307 (2013)], we could explain these behaviors by introducing a model of supercooled liquids in which local rearrangements interact via elasticity. However, in that model the disorder char…
Stochastic-Beta-Relaxation (SBR) provides a characterisation of the glass crossover in discontinuous Spin-Glasses and Supercoooled liquid. Notably it can be derived through a rigorous computation from a dynamical Landau theory. In this paper I will discuss the precise meaning of this connection in a language that does not require familiarity with statistical field theory. I will discuss finite-size corrections in mean-field Spin-Glass models and loop corrections in finite-dimensional models that are both described by the dynamical Landau theory considered. Then I will argue that the same Landa…
A statistical analysis of the geometries of particle trajectories in the supercooled liquid state is reported. We examine two structurally different fragile glass-forming liquids simulated by molecular dynamics. In both liquids, the trajectories are found to exhibit a long-range localisation distinct from the short-range localisation within the cage of nearest neighbours. This novel diffusion anomaly is interpreted as a result of the potential-energy landscape topography of fragile glass-formers where the local energy minima coalesce into metabasins - compact domains with low escape probabilit…
We study the transport and the relaxation properties of a molecular supercooled liquid by molecular-dynamics numerical simulations. The focus is on the translational motion. Jump motion is detected. At lower temperature the Stokes-Einstein law breaks down.
It is proposed that the rate of relaxation in a liquid is better described by the geometric mean of the van Hove distribution function, rather than the standard arithmetic mean used to obtain the mean squared displacement. The difference between the two means is shown to increase significantly with an increase in the non-Gaussian character of the displacement distribution. Preliminary results indicate that the geometric diffusion constant results in a substantial reduction of the deviation from Stokes-Einstein scaling.
Reply to "Comment on 'Fickian non-Gaussian diffusion in glass-forming liquids' ". In [ArXiv:2210.07119v1], Berthier et al. questioned the findings of our letter [Phys. Rev. Lett. 128, 168001 (2022)], concerning the existence and the features of Fickian non-Gaussian diffusion in glass-forming liquids. Here we demonstrate that their arguments are either wrong, or not meaningful to our scope. Thus, we fully confirm the validity and novelty of our results.
A model for the supercooled liquid is considered by taking into account its solid like properties. We focus on how the long time dynamics is affected due to the coupling between the slowly decaying density fluctuations and the local displacement variables in the frozen liquid. Results from our model agree with the recent observation of Novikov and Sokolov [Nature (London) {\bf 431}, 961 (2004)] that the fragility index $m$ of a glass forming material is linearly related to the corresponding ratio $K/G$ of the bulk and the shear moduli.
The rotational dynamics of a supercooled molecular liquid is investigated by a molecular-dynamics numerical study. We detect rotational jumps with a waiting-time distribution which is well fitted by a truncated power law. At lower temperatures the rotational correlation times are decoupled from the viscosity.
The existence of a phase transition between two distinct liquid phases in single-component network-forming liquids (e.g., water, silica, silicon) has elicited considerable scientific interest. The challenge, both for experiments and simulations, is that the liquid-liquid phase transition occurs under deeply supercooled conditions, where crystallization occurs very rapidly. Thus, early evidence from numerical equation of state studies was challenged, with the argument that slow spontaneous crystallization had been misinterpreted as evidence of a second liquid state. Rigorous free energy calcula…
Below the melting temperature $T_m$ crystals are the stable phase of typical elemental or molecular systems. However, cooling down a liquid below $T_m$, crystallization is anything but inevitable. The liquid can be supercooled, eventually forming a glass below the glass transition temperature $T_g$. Despite their long lifetimes and the presence of strong barriers that produces an apparent stability, supercooled liquids and glasses remain intrinsically metastable state and thermodynamically unstable towards the crystal. Here we investigated the isothermal crystallization kinetics of the prototy…
Because of the isotropic and disordered nature of liquids, the anisotropy hidden in intermolecular interactions are often neglected. Accordingly, the order parameter describing a simple liquid has so far been believed to be only density. In contrast to this common sense, we propose that two order parameters, namely, density and bond order parameters, are required to describe the phase behavior of liquids since they intrinsically tend to form local bonds. This model gives us clear physical explanations for two poorly-understood phenomena in supercooled liquids: (i) large-scale density fluctuati…
We investigate to what extend the replica-exchange Monte Carlo method is able to equilibrate a simple liquid in its supercooled state. We find that this method does indeed allow to generate accurately the canonical distribution function even at low temperatures and that its efficiency is about 10-100 times higher than the usual canonical molecular dynamics simulation.
We investigate the properties of water along the liquid/vapor coexistence line in the supercooled regime down to the no-man's land. Extensive molecular dynamics simulations of the TIP4P/2005 liquid/vapor interface in the range 198 -- 348 K allow us to locate the second surface tension inflection point with high accuracy at 283$\pm$5 K, close to the temperature of maximum density (TMD). This temperature also coincides with the appearance of a density anomaly at the interface known as the apophysis. We relate the emergence of the apophysis to the observation of HDL water adsorption in the proxim…
Molecular dynamics simulations are performed for a supercooled simple liquid with changing the system size from N=108 to $10^4$ to examine possible finite-size effects. Although almost no systematic deviation is detected in the static pair correlation functions, it is demonstrated that the structural $α$ relaxation in a small system becomes considerably slower than that in larger systems for temperatures below $T_c$ at which the size of the cooperative particle motions becomes comparable to the unit cell length of the small system. The discrepancy increases with decreasing temperature.
We perform molecular-dynamics simulations of a molecular system in supercooled states for different values of inertia parameters to provide evidence that the long-time dynamics depends only on the equilibrium structure. This observation is consistent with the prediction of the mode-coupling theory for the glass transition and with the hypothesis that the potential energy-landscape controls the slow dynamics. We also find that dynamical properties at intermediate wavenumber depend on the spatial correlation of the molecule's geometrical center.
One of the hallmarks of molecular dynamics in deeply supercooled liquids is the non-exponential character of the relaxation functions. It has been a long standing issue if 'universal' features govern the lineshape of glassy dynamics independent of any particular molecular structure or interaction. In the paper, we elucidate this matter by giving a comprehensive comparison of the spectral shape of depolarized light scattering and dielectric data of deeply supercooled liquids. The light scattering spectra of very different systems, e.g. hydrogen bonding and van der Waals liquids but also ionic s…
We present mode-coupling equations for the description of the slow dynamics observed in supercooled molecular liquids close to the glass transition. The mode-coupling theory (MCT) originally formulated to study the slow relaxation in simple atomic liquids, and then extended to the analysis of liquids composed by linear molecules, is here generalized to systems of arbitrarily shaped, rigid molecules. We compare the predictions of the theory for the $q$-vector dependence of the molecular nonergodicity parameters, calculated by solving numerically the molecular MCT equations in two different appr…
We have performed molecular dynamics simulations for a model two-dimensional soft-core mixture in a supercooled state. The mixture exhibits a slow structural relaxation in a quiescent state, however, the relaxation is much enhanced in sheared states. There observed surprisingly small anisotropy both in the coherent and incoherent density correlation functions even under extremely strong shear which is $10^3$ times faster than the structural relaxation rate. The present simulation results agree well with predictions of the recently developed mode-coupling theory in shear.
We determine the phase diagram of liquid supercooled water by extensive computer simulations using the TIP5P-E model [J. Chem. Phys. {\bf 120}, 6085 (2004)]. We find that the transformation of water into a low density liquid in the supercooled range strongly enhances the solubility of hydrophobic particles. The transformation of water into a tetrahedrally structured liquid is accompanied by a minimum in the hydration entropy and enthalpy. The corresponding change in sign of the solvation heat capacity indicates a loss of one characteristic signature of hydrophobic hydration. The observed behav…
The effect of temperature on friction and slip at the liquid-solid interface has attracted attention over the last twenty years, both numerically and experimentally. However, the role of temperature on slip close to the glass transition has been less explored. Here, we use molecular dynamics to simulate a bi-disperse atomic fluid, which can remain liquid below its melting point (supercooled state), to study the effect of temperature on friction and slip length between the liquid and a smooth apolar wall, in a broad range of temperatures. At high temperatures, an Arrhenius law fits well the tem…
We study the effect of heavy impurities on the dynamics of supercooled liquids. When a small fraction of particles in the supercooled liquid is made heavier, they exhibit slower dynamics than the original particles and also make the overall system slower. If one looks at the overlap correlation function to quantify dynamics in the system, it has different behavior for the heavy and the light particles. In particular, at the relaxation time of the overall system, the degree of relaxation achieved by the heavier particles is lesser on average than that achieved by the light particles. This diffe…
We identify the pattern of microscopic dynamical relaxation for a two dimensional glass forming liquid. On short timescales, bursts of irreversible particle motion, called cage jumps, aggregate into clusters. On larger time scales, clusters aggregate both spatially and temporally into avalanches. This propagation of mobility, or dynamic facilitation, takes place along the soft regions of the systems, which have been identified by computing isoconfigurational Debye-Waller maps. Our results characterize the way in which dynamical heterogeneity evolves in moderately supercooled liquids and reveal…
We use a recently-derived reformulation of the diffusion constant [Stillinger F H and Debenedetti P G 2005 J. Phys. Chem. B 109 6604] to investigate heterogeneous dynamics and non-Gaussian diffusion in a binary Lennard-Jones mixture. Our work focuses on the joint probability distribution of particles with velocity v_0 at time t=0 and eventual displacement Delta x at time t=Delta t. We show that this distribution attains a distinctive shape at the time of maximum non-Gaussian behavior in the supercooled liquid. By performing a two-Gaussian fit of the displacement data, we obtain, in a non-arbit…
The supercooled state of bulk water is largely hidden by unavoidable crystallization, which creates an experimentally inaccessible temperature regime - a 'no man's land'. We address this and circumvent the crystallization problem by systematically studying the supercooled dynamics of hydrogen bonded oligomeric liquids (glycols), where water corresponds to the chain-ends alone. This novel approach permits a 'dilution of water' by altering the hydrogen bond concentration via variations in chain length. We observe a dynamic crossover in the temperature dependence of the structural relaxation time…
A critique of a recent paper [Mallamace et al PNAS 2010 107 (52) 22457-2246] purporting the existence of a fragile-to-strong crossover in super cooled liquids is presented. It is shown that Mallamace et al have mistaken the onset crossover with the fragile-strong crossover. Moreover, it is shown that supercooled liquid transport data follows the expected parabolic function, without a crossover to another functional form, and that Mallamace et al have obscured this fact by singling out anomalous data as examples.
Using a novel thermodynamic integration scheme, we compute the excess free energy, $γ$, of a glass-forming, binary Lennard-Jones liquid in contact with a frozen amorphous wall, formed by particles frozen into a similar structure as the liquid. We find that $γ$ is non-zero, becoming negative at low temperature. This indicates that the thermodynamics of the system is perturbed by the effect of the amorphous wall.
We solve numerically the equations of nonlinear fluctuating hydrodynamics (NFH) for the supercooled liquid. The time correlation of the density fluctuations in equilibrium obtained here shows quantitative agreement with molecular dynamics(MD) simulation data. We demonstrate numerically that the 1/rho nonlinearity in the NFH equations of motion is essential in restoring the ergodic behavior in the liquid. Under nonequilibrium conditions the time correlation functions relax in a manner similar to that observed in the molecular dynamics simulations in binary mixtures. The waiting time t_w depende…
We demonstrate, via numerical simulations, that the relaxation dynamics of supercooled liquids correlates well with a plastic length scale measuring a particle's response to impulsive localized perturbations and weakly to measures of local elasticity. We find that the particle averaged plastic length scale vanishes linearly in temperature and controls the super-Arrhenius temperature dependence of the relaxation time. Furthermore, we show that the plastic length scale of individual particles correlates with their typical displacement at the relaxation time. In contrast, the local elastic respon…
A theoretical treatment of deeply supercooled liquids is difficult because their properties emerge from spatial inhomogeneities that are self-induced, transient, and nanoscopic. I use computer simulations to analyse self-induced static and dynamic heterogeneity in equilibrium systems approaching the experimental glass transition. I characterise the broad sample-to-sample fluctuations of salient dynamic and thermodynamic properties in elementary mesoscopic systems. Findings regarding local lifetimes and distributions of dynamic heterogeneity are in excellent agreement with recent single molecul…
A metastable state, characterized by a low degree of mass localization is identified using Density Functional Theory. This free energy minimum, located through the proper evaluation of the competing terms in the free energy functional, is independent of the specific form of the DFT used. Computer simulation results on particle motion indicate that this heterogeneous state corresponds to the supercooled state.
A popular hypothesis that explains the anomalies of supercooled water is the existence of a metastable liquid-liquid transition hidden below the line of homogeneous nucleation. If this transition exists and if it is terminated by a critical point, the addition of a solute should generate a line of liquid-liquid critical points emanating from the critical point of pure metastable water. We have analyzed thermodynamic consequences of this scenario. In particular, we consider the behavior of two systems, H2O-NaCl and H2O-glycerol. We find the behavior of the heat capacity in supercooled aqueous s…
Liquid-liquid and liquid-vapor coexistence regions of various water models were determined by MC simulations of isotherms of density fluctuation restricted systems and by Gibbs ensemble MC simulations. All studied water models show multiple liquid-liquid phase transitions in the supercooled region: we observe two transitions of the TIP4P, TIP5P and SPCE model and three transitions of the ST2 model. The location of these phase transitions with respect to the liquid-vapor coexistence curve and the glass temperature is highly sensitive to the water model and its implementation. We suggest, that t…
The primary and secondary relaxation timescales of aging colloidal suspensions of Laponite are estimated from intensity autocorrelation functions obtained in dynamic light scattering (DLS) experiments. The dynamical slowing down of these relaxation processes are compared with observations in fragile supercooled liquids by establishing a one-to-one mapping between the waiting time since filtration of a Laponite suspension and the inverse of the temperature of a supercooled liquid that is rapidly quenched towards its glass transition temperature. New timescales, such as the Vogel time and the Ka…
