We have investigated the validity of the fluctuation-dissipation theorem (FDT) and the applicability of the concept of effective temperature in a number of non-equilibrium soft glassy materials. Using a combination of passive and active microrheology to measure displacement fluctuations and the mechanical response function of probe particles embedded in the materials, we have directly tested the validity of the FDT. Our results show no violation of the FDT over several decades in frequency (1-10$^4$ Hz) for hard sphere colloidal glasses and colloidal glasses and gels of Laponite. We further extended the bandwidth of our measurements to lower frequencies (down to 0.1 Hz) using video microscopy to measure the displacement fluctuations, again without finding any deviations from the FDT.
In a glassy system different degrees of freedom have well-separated characteristic times, and are described by different temperatures. The stationary state is essentially non-equilibrium. A generalized statistical thermodynamics is constructed and a universal variational principle is proposed. Entropy production and energy dissipation occur at a constant rate; there exists a universal relation between them, valid to leading order in the small ratio of the characteristic times. Energy dissipation (unlike entropy production) is closely connected to the fluctuations of the slow degree. Correction…
A method is presented for measuring the integrated response in Ising spin system without applying any perturbing field. Large-scale simulations are performed in order to show how the method works. Very precise measurements of the fluctuation-dissipation ratio are presented for 3 different Ising models: the 2-dimensional ferromagnetic model, the mean-field diluted 3-spin model, and the 3-dimensional Edwards-Anderson model.
We present a version of soft glassy rheology that includes thermalized strain degrees of freedom. It fully specifies systems' strain-history-dependent positions on their energy landscapes and therefore allows for quantitative analysis of their heterogeneous yielding dynamics and nonequilibrium deformation thermodynamics. As a demonstration of the method, we illustrate the very different characteristics of fully-thermal and nearly-athermal plasticity by comparing results for thermalized and nonthermalized plastic flow.
The scaling properties of the soft-sphere potential allow the derivation of an exact expression for the pressure of a frozen liquid, i.e., the pressure corresponding to configurations which are local minima in its multidimensional potential energy landscape. The existence of such a relation offers the unique possibility for testing the recently proposed extension of the liquid free energy to glassy out-of-equilibrium conditions and the associated expression for the temperature of the configurational degrees of freedom. We demonstrate that the non-equilibrium free energy provides an exact descr…
The rheological behavior of soft glassy materials basically results from the interplay between shearing forces and an intrinsic slow dynamics. This competition can be described by a microscopic theory, which can be viewed as a nonequilibrium schematic mode-coupling theory. This statistical mechanics approach to rheology results in a series of detailed theoretical predictions, some of which still awaiting for their experimental verification. We present new, preliminary, results about the description of yield stress, flow heterogeneities and activated processes within this theoretical framework.
We analytically study a one dimensional compaction model in the glassy regime. Both correlation and response functions are calculated exactly in the evolving dense and low tapping strength limit, where the density relaxes in a $1/\ln t$ fashion. The response and correlation functions turn out to be connected through a non-equilibrium generalisation of the fluctuation-dissipation theorem. The initial response in the average density to an increase in the tapping strength is shown to be negative, while on longer timescales it is shown to be positive. On short time scales the fluctuation-dissipati…
We study the emergence of critical dynamics in the steady shear rheology of fluidized soft glassy materials. Within a mesoscale elasto-plastic model accounting for a shear band instability, we show how an additional noise can induce a transition from phase separated to homogeneous flow, accompanied by critical-like fluctuations of the macroscopic shear rate. Both macroscopic quantities and fluctuations exhibit power law behaviors in the vicinity of this transition, consistent with previous experimental findings on vibrated granular media. Altogether, our results suggest a generic scenario for …
The autocorrelation function of pattern fluctuation is used to study soft-mode turbulence (SMT), a spatiotemporal chaos observed in homeotropic nematics. We show that relaxation near the electroconvection threshold deviates from the exponential. To describe this relaxation, we propose a compressed exponential appearing in dynamics of glass forming liquids. Our findings suggest that coherent motion contributes to SMT dynamics. We also confirmed that characteristic time is inversely proportional to electroconvection's control parameter.
Single cells often generate precise responses by involving dissipative out-of-thermodynamic equilibrium processes in signaling networks. The available free energy to fuel these processes could become limited depending on the metabolic state of an individual cell. How does limiting dissipation affect the kinetics of high precision responses in single cells? I address this question in the context of a kinetic proofreading scheme used in a simple model of early time T cell signaling. I show using exact analytical calculations and numerical simulations that limiting dissipation qualitatively chang…
In the free energy landscape picture of glassy systems, the slow dynamics characteristic of these systems is believed to be due to the existence of a complicated free-energy landscape with many local minima. We show here that for a colloidal glassy material multiple paths can be taken through the free energy landscape, that can even lead to different 'final' non-ergodic states at the late stages of aging. We provide clear experimental evidence for the distinction of gel and glassy states in the system and show that for a range of colloid concentrations, the transition to non-ergodicity can occ…
Emergent odd viscoelasticity in chiral soft glassy materials
University of Göttingen（哥廷根大学）；Ecole Polytechnique de l’Université de Nantes, LTeN, U6607, Nantes Université（南特大学理工学院，LTeN，U6607）；King’s College London（伦敦国王学院）
The `soft glassy rheology' (SGR) model gives an appealing account of the flow of nonergodic soft materials in terms of the local yield dynamics of mesoscopic elements. Newtonian, power-law, and yield-stress fluid regimes arise on varying a `noise temperature', x. Here we extend the model, to capture the idea that the noise is largely caused by yield itself. The extended model can account for the viscosity-bifurcation and shear-banding effects reported recently in a wide range of soft materials. A variant model may shed light on shear banding and strain-rate hysteresis seen in glassy star polym…
We analyze fluctuation-dissipation relations in the Backgammon model: a system that displays glassy behavior at zero temperature due to the existence of entropy barriers. We study local and global fluctuation relations for the different observables in the model. For the case of a global perturbation we find a unique negative fluctuation-dissipation ratio that is independent of the observable and which diverges linearly with the waiting time. This result suggests that a negative effective temperature can be observed in glassy systems even in the absence of thermally activated processes.
We combine the hyper-netted chain approximation of liquid state theory with the mode-coupling theory of the glass transition to analyze the structure and dynamics of soft spheres interacting via harmonic repulsion. We determine the locus of the fluid-glass dynamic transition in a temperature -- volume fraction phase diagram. The zero-temperature (hard sphere) glass transition influences the dynamics at finite temperatures in its vicinity. This directly implies a form of dynamic scaling for both the average relaxation time and dynamic susceptibilities quantifying dynamic heterogeneity. We discu…
Solids deform and fluids flow, but soft glassy materials, such as emulsions, foams, suspensions, and pastes, exhibit an intricate mix of solid and liquid-like behavior. While much progress has been made to understand their elastic (small strain) and flow (infinite strain) properties, such understanding is lacking for the softening and yielding phenomena that connect these asymptotic regimes. Here we present a comprehensive framework for softening and yielding of soft glassy materials, based on extensive numerical simulations of oscillatory rheological tests, and show that two distinct scenario…
Violation of the fluctuation-dissipation theorem (FDT) in a frustrated Heisenberg model on the Kagome lattice is investigated using Monte Carlo simulations. The model exhibits glassy behaviour at low temperatures accompanied by very slow dynamics. Both the spin-spin autocorrelation function and the response to an external magnetic field are studied. Clear evidence of a constant value of the fluctuation dissipation ratio and long range memory effects are observed for the first time in this model. The breakdown of the FDT in the glassy phase follows the predictions of the mean field theory for s…
We study numerically the Kawasaki dynamics of the 2d Ising model. At large time we recover the coarsening growth as l(t) ~ t^(1/3). At shorter time however, the system enters a metastable glassy regime that displays an extremely slow growth and non-trivial violations of the fluctuation dissipation theorem similar to those observed in spin glasses: this is one of the simplest system where such violations occur. We also consider Potts models, where a similar behavior is observed, and the modelof Shore and Sethna where the domain growth is also slow, but where violations of the fluctuation dissip…
We show that a generalized fluctuation-dissipation relation applies upon instantaneously increasing the temperature of a deeply supercooled liquid. This has the same two-step shape of the relation found upon cooling the liquid, but with opposite violation, indicating an effective temperature that is lower than bath temperature. We show that the effective temperature exhibits some sensible time-dependence and that it retains its connection with the partitioned phase space visited in ageing. We underline the potential relevance of our numerical results for experimental studies of the fluctuation…
There is a growing evidence that relaxation in glassy materials, both spontaneous and externally driven, is mediated by localized soft spots. Recent progress made it possible to identify the soft spots inside glassy structures and to quantify their degree of softness. These softness measures, however, are typically scalars, not taking into account the tensorial/anisotropic nature of soft spots, which implies orientation-dependent coupling to external deformation. Here we derive from first principles the linear response coupling between the local heat capacity of glasses, previously shown to pr…
The predictions of a nonequilibrium schematic mode-coupling theory developed to describe the nonlinear rheology of soft glassy materials have been numerically challenged in a sheared binary Lennard-Jones mixture. The theory gives an excellent description of the stress/temperature `jamming phase diagram' of the system. In the present paper, we focus on the issue of an effective temperature Teff for the slow modes of the fluid, as defined from a generalized fluctuation-dissipation theorem. As predicted theoretically, many different observables are found to lead to the same value of Teff, suggest…
We analyze the validity of the fluctuation-dissipation theorem for slow relaxation systems in the context of mesoscopic nonequilibrium thermodynamics. We demonstrate that the violation arises as a natural consequence of the elimination of fast variables in the description of a glassy system, and it is intrinsically related to the underlying activated nature of slow relaxation. In addition, we show that the concept of effective temperature, introduced to characterize the magnitude of the violation, is not robust since it is observable-dependent, can diverge, or even be negative.
A picture for thermodynamics of the glassy state is introduced. It assumes that one extra parameter, the effective temperature, is needed to describe the glassy state. This explains the classical paradoxes concerning the Ehrenfest relations and the Prigogine-Defay ratio. As a second part, the approach connects the response of macroscopic observables to a field change with their temporal fluctuations, and with the fluctuation-dissipation relation, in a generalized non-equilibrium way.
A picture for thermodynamics of the glassy state is introduced. It assumes that one extra parameter, the effective temperature, is needed to describe the glassy state. This explains the classical paradoxes concerning the Ehrenfest relations and the Prigogine-Defay ratio. As a second part, the approach connects the response of macroscopic observables to a field change with their temporal fluctuations, and with the fluctuation-dissipation relation, in a generalized non-equilibrium way.
Single and double torsion oscillators have been used successfully to measure the anomalous change in resonant frequency and accompanying dissipation in solid He-4. We present a glass description of the mechanical anomalies found in torsion oscillator measurements. Our results show that it is not necessary to invoke a supersolid interpretation to explain these mechanical anomalies. Previously, we demonstrated that the back-action of a glassy subsystem present in solid He-4 can account for frequency change and dissipation peak in many single torsion oscillator experiments. Here, we show that the…
Mesoscopic models play an important role in our understanding of the deformation and flow of amorphous materials. One such description, based on the Shear Transformation Zone (STZ) theory, has recently been re-formulated within a non-equilibrium thermodynamics framework, and found to be consistent with it. We show here that a similar interpretation can be made for the Soft Glassy Rheology (SGR) model. Conceptually this means that the "noise temperature" x, proposed phenomenologically in the SGR model to control the dynamics of a set of slow mesoscopic degrees of freedom, can consistently be in…
We investigate by Montecarlo simulation the linear response function of three dimensional structural glass models defined by short-range kinetic constraints and a trivial equilibrium Boltzmann-Gibbs measure. The breakdown of the fluctuation-dissipation theorem in the glassy phase follows the prediction of mean field low temperature mode-coupling theory.
A system is glassy when the observation time is much smaller than the equilibration time. A unifying thermodynamic picture of the glassy state is presented. Slow configurational modes are in quasi-equilibrium at an effective temperature. It enters thermodynamic relations with the configurational entropy as conjugate variable. Slow fluctuations contribute to susceptibilities via quasi-equilibrium relations, while there is also a configurational term. Fluctuation-dissipation relations also involve the effective temperature. Fluctuations in the energy are non-universal, however. The picture is su…
We show the existence of intermittent dynamics in one of the simplest model of a glassy system: the two-state model, which has been used to explain the origin of the violation of the fluctuation-dissipation theorem. The dynamics is analyzed through a Langevin equation for the evolution of the state of the system through its energy landscape. The results obtained concerning the violation factor and the non-Gaussian nature of the fluctuations are in good qualitative agreement with experiments measuring the effective temperature and the voltage fluctuations in gels and in polymer glasses. The met…
The nonequilibrium dynamics of a binary Lennard-Jones mixture in a simple shear flow is investigated by means of molecular dynamics simulations. The range of temperature investigated covers both the liquid, supercooled and glassy states, while the shear rate covers both the linear and nonlinear regimes of rheology. The results can be interpreted in the context of a nonequilibrium, schematic mode-coupling theory developed recently, which makes the theory applicable to a wide range of soft glassy materials. The behavior of the viscosity is first investigated. In the nonlinear regime, strong shea…
Many amorphous materials show spatially heterogenous dynamics, as different regions of the same system relax at different rates. Such a signature, known as Dynamic Heterogeneity, has been crucial to understand the jamming transition in simple model systems and, currently, is considered very promising to characterize more complex fluids of industrial and biological relevance. Unfortunately, measurements of dynamic heterogeneities typically require sophysticated experimental set-ups and are performed by few specialized groups. It is now possible to quantitatively characterize the relaxation proc…
While the glass transition at non-zero temperature seems to be hard to access for experimental, theoretical, or simulation studies, jamming at zero temperature has been explored in great detail. It is a widely discussed question whether this athermal jamming transition is related to the glass transition. Motivated by the exploration of the energy landscape that has been successfully used to describe athermal jamming, we introduce a new method to determine whether the configuration space of a soft sphere system can be explored within a reasonable timescale or not, i.e., whether the system is er…
We introduce a lattice spin model where frustration is due to multibody interactions rather than quenched disorder in the Hamiltonian. The system has a crystalline ground state and below the melting temperature displays a dynamic behaviour typical of fragile glasses. However, the supercooled phase loses stability at an effective spinodal temperature, and thanks to this the Kauzmann paradox is resolved. Below the spinodal the system enters an off-equilibrium regime corresponding to fast crystal nucleation followed by slow activated crystal growth. In this phase and in a time region which is lon…
We study the intermittent behavior of the energy decay and linear magnetic response of a glassy system during isothermal aging after a deep thermal quench using the Edward-Anderson spin glass model as a paradigmatic example. The large intermittent changes in the two observables are found to occur in a correlated fashion and through irreversible bursts, `quakes', which punctuate reversible and equilibrium-like fluctuations of zero average. The temporal distribution of the quakes it found to be a Poisson distribution with an average growing logarithmically on time, indicating that the quakes are…
In recent years, the paradigm of `soft glassy matter' has been used to describe diverse nonergodic materials exhibiting strong local disorder and slow mesoscopic rearrangement. As so far formulated, however, the resulting `soft glassy rheology' (SGR) model treats the shear stress in isolation, effectively `scalarizing' the stress and strain rate tensors. Here we offer generalizations of the SGR model that combine its nontrivial aging and yield properties with a tensorial structure that can be specifically adapted, for example, to the description of fluid film assemblies or disordered foams.
We study numerically the out-of-equilibrium dynamics of interfaces at finite temperatures when driven well below the zero-temperature depinning threshold. We go further than previous analysis by including the most relevant non-equilibrium correction to the elastic Hamiltonian. We find that the relaxation dynamics towards the steady-state shows glassy behavior, aging and violation of the fluctuation-dissipation theorem. The interface roughness exponent alpha approx 0.7 is found to be robust to temperature changes. We also study the instantaneous velocity signal in the low temperature regime and…
The existence of a generalized fluctuation-dissipation theorem observed in simulations and experiments performed in various glassy materials is related to the concepts of local equilibration and heterogeneity in space. Assuming the existence of a dynamic coherence length scale up to which the system is locally equilibrated, we extend previous generalizations of the FDT relating static to dynamic quantities to the physically relevant domain where asymptotic limits of large times and sizes are not reached. The formulation relies on a simple scaling argument and has thus not the character of a th…
Structural glasses formed by quenching a melt are known to host a population of low-energy quasilocalized (nonphononic) excitations whose frequencies $ω$ follow a universal $\sim\!ω^4$ distribution as $ω\!\to\!0$, independently of the glass formation history, the interparticle interaction potential or spatial dimension. Here, we show that the universal quartic law of nonphononic excitations also holds in disordered crystals featuring finite long-range order, which is absent in their glassy counterparts. We thus establish that the degree of universality of the quartic law extends beyond structu…
In glassy materials aging proceeds at large times via thermal activation. We show that this can lead to negative dynamical response functions and novel and well-defined violations of the fluctuation-dissipation theorem, in particular, negative fluctuation-dissipation ratios. Our analysis is based on detailed theoretical and numerical results for the activated aging regime of simple kinetically constrained models. The results are relevant to a variety of physical situations such as aging in glass-formers, thermally activated domain growth and granular compaction.
For soft matter systems strongly driven by stationary flow, we discuss an extended fluctuation-dissipation theorem (FDT). Beyond the linear response regime, the FDT for the stress acquires an additional contribution involving the observable that is conjugate to the strain rate with respect to the dissipation function. This extended FDT is evaluated both analytically for Rouse polymers and in numerical simulations for colloidal suspensions. More generally, our results suggest an extension of Onsager's regression principle to nonequilibrium steady states.
We present the study of the landscape structure of athermal soft spheres both as a function of the packing fraction and of the energy. We find that, on approaching the jamming transition, the number of different configurations available to the system has a steep increase and that a hierarchical organization of the landscape emerges. We use the knowledge of the structure of the landscape to predict the values of thermodynamic observables on the edge of the transition.
A picture for thermodynamics of the glassy state was introduced recently by us (Phys. Rev. Lett. {\bf 79} (1997) 1317; {\bf 80} (1998) 5580). It starts by assuming that one extra parameter, the effective temperature, is needed to describe the glassy state. This approach connects responses of macroscopic observables to a field change with their temporal fluctuations, and with the fluctuation-dissipation relation, in a generalized, non-equilibrium way. Similar universal relations do not hold between energy fluctuations and the specific heat. In the present paper the underlying arguments are disc…
In this paper, we proved that by choosing the proper variational function and variables, the variational approach proposed by M. Doi in soft matter physics was equivalent to the Conservation-Dissipation Formalism. To illustrate the correspondence between these two theories, several novel examples in soft matter physics, including particle diffusion in dilute solutions, polymer phase separation dynamics and nematic liquid crystal flows, were carefully examined. Based on our work, a deep connection among the generalized Gibbs relation, the second law of thermodynamics and the variational princip…
The physics of glasses can be studied from many viewpoints, from material scientists interested in the development of new materials to statistical physicists inventing new theoretical tools to deal with disordered systems. In these lectures I described a variety of physical phenomena observed in actual glassy materials, from disordered magnetic systems to soft gels. Despite the very large gap between experimental and numerical time windows, I showed that computer simulations represent an efficient theoretical tool which can shed light on the microscopic origins of glassy dynamics.
The dynamics of a dense binary mixture of soft dumbbells, each subject to an active propulsion force and thermal fluctuations, shows a sudden arrest, first to a translational then to a rotational glass, as one reduces temperature $T$ or the self-propulsion force $f$. Is the temperature-induced glass different from the activity-induced glass? To address this question, we monitor the dynamics along an iso-relaxation-time contour in the $(T-f)$ plane. We find dramatic differences both in the fragility and in the nature of dynamical heterogeneity which characterise the onset of glass formation - t…
We solve exactly and describe in detail a simplified scalar model for the low frequency shear rheology of foams, emulsions, slurries, etc. [P. Sollich, F. Lequeux, P. Hebraud, M.E. Cates, Phys. Rev. Lett. 78, 2020 (1997)]. The model attributes similarities in the rheology of such ``soft glassy materials'' to the shared features of structural disorder and metastability. By focusing on the dynamics of mesoscopic elements, it retains a generic character. Interactions are represented by a mean-field noise temperature x, with a glass transition occurring at x=1 (in appropriate units). The exact sol…
We study spatio-temporal fluctuations in the non-equilibrium dynamics of the d dimensional O(N) in the large N limit. We analyse the invariance of the dynamic equations for the global correlation and response in the slow ageing regime under transformations of time. We find that these equations are invariant under scale transformations. We extend this study to the action in the dynamic generating functional finding similar results. This model therefore falls into a different category from glassy problems in which full time-reparametrisation invariance, a larger symmetry that emcompasses time sc…
We develop a new analytic approach for the study of lattice heteropolymers, and apply it to copolymers with correlated Markovian sequences. According to our analysis, heteropolymers present three different dense phases depending upon the temperature, the nature of the monomer interactions, and the sequence correlations: (i) a liquid phase, (ii) a ``soft glass'' phase, and (iii) a ``frozen glass'' phase. The presence of the new intermediate ``soft glass'' phase is predicted for instance in the case of polyampholytes with sequences that favor the alternation of monomers. Our approach is based on…
The validity of fluctuation dissipation relations in an aging system is studied in a colloidal glass during the transition from a fluid-like to a solid-like state. The evolution of the rheological and electrical properties is analyzed in the range $1Hz - 40Hz$. It is found that at the beginning of the transition the fluctuation dissipation relation is strongly violated in electrical measurements. The amplitude and the persistence time of this violation are decreasing functions of frequency. At the lowest frequencies of the measuring range it persists for times which are about 5% of the time ne…
We give a brief review of violations of the fluctuation-dissipation theorem (FDT) in out-of-equilibrium systems; in mean field scenarios the corresponding fluctuation-dissipation (FD) plots can, in the limit of long times, be used to define a {\em effective temperature} $\teff$ that shares many properties of the true thermodynamic temperature $T$. We discuss carefully how correlation and response functions need to be represented to obtain meaningful limiting FD plots in {\em non-mean field} systems. A minimum requirement on the resulting effective temperatures is that they should be independen…
A unified derivation of the off equilibrium fluctuation dissipation relations (FDR) is given for Ising and continous spins to arbitrary order, within the framework of Markovian stochastic dynamics. Knowledge of the FDR allows to develop zero field algorithms for the efficient numerical computation of the response functions. Two applications are presented. In the first one, the problem of probing for the existence of a growing cooperative length scale is considered in those cases, like in glassy systems, where the linear FDR is of no use. The effectiveness of an appropriate second order FDR is …
Entropy and the fluctuation-dissipation theorem are at the heart of statistical mechanics near equilibrium. Driving a system beyond the linear response regime leads to (i) the breakdown of the fluctuation-dissipation theorem and (ii) a nonzero entropy production rate. We show how both phenomena are related using the general framework of stochastic thermodynamics suitable for soft matter systems governed by stochastic dynamics and driven through nonconservative forces or external flows. In particular, the excess of the fluctuation-dissipation theorem in a nonequilibrium steady state compared to…
We study self-diffusion within a simple hopping model for glassy materials. (The model is Bouchaud's model of glasses [J.-P. Bouchaud, J. Physique I 2, 1705 (1992)], as extended to describe rheological properties [P. Sollich, F. Lequeux, P. Hebraud and M.E. Cates, Phys. Rev. Lett. 78, 2020 (1997)].) We investigate the breakdown, near the glass transition, of the (generalized) Stokes-Einstein relation between self-diffusion of a tracer particle and the (frequency-dependent) viscosity of the system as a whole. This stems from the presence of a broad distribution of relaxation times of which diff…
We report the results of molecular dynamics simulations of the protein myosin carried out with an elastic network model. Quenching the system, we observe glassy behavior of a density correlation function and a density response function that are often investigated in structure glasses and spin glasses. In the equilibrium, the fluctuation-response relation, a representative relation of the fluctuation-dissipation theorem, holds that the ratio of the density correlation function to the density response function is equal to the temperature of the environment. We show that in the quenched system th…
We review a statistical picture of the glassy state derived from the analysis of the off-equilibrium fluctuation-dissipation relations. We define an ultra-long time limit where ``one time quantities'' are close to equilibrium while response and correlation can still display aging. In this limit it is possible to relate the fluctuation-response relation to static breaking of ergodicity. The resulting picture suggests that even far from that limit, the fluctuation-dissipation ratio relates to the rate of growth of the configurational entropy with free-energy density.
We consider the dynamics of a diluted mean-field spin glass model in the aging regime. The model presents a particularly rich heterogeneous behavior. In order to catch this behavior, we perform a **spin-by-spin analysis** for a **given disorder realization**. The results compare well with the outcome of a static calculation which uses the ``survey propagation'' algorithm of Mezard, Parisi, and Zecchina [Sciencexpress 10.1126/science.1073287 (2002)]. We thus confirm the connection between statics and dynamics at the level of single degrees of freedom. Moreover, working with single-site quantiti…
