Laboratory studies play a crucial role in understanding the chemical nature of the interstellar medium (ISM), but the disconnect between experimental timescales and the timescales of reactions in space can make a direct comparison between observations, laboratory, and model results difficult. Here we study the survival of reactive fragments of the polycyclic aromatic hydrocarbon (PAH) coronene, where individual C atoms have been knocked out of the molecules in hard collisions with He atoms at stellar wind and supernova shockwave velocities. Ionic fragments are stored in the DESIREE cryogenic ion-beam storage ring where we investigate their decay for up to one second. After 10 ms the initially hot stored ions have cooled enough so that spontaneous dissociation no longer takes place at a measurable rate; a majority of the fragments remain intact and will continue to do so indefinitely in isolation. Our findings show that defective PAHs formed in energetic collisions with heavy particles may survive at thermal equilibrium in the interstellar medium indefinitely, and could play an important role in the chemistry in there, due to their increased reactivity compared to intact or photo-fragmented PAHs. Ion storage rings allow reactions to be studied over orders of magnitude in time, bridging the gap between typical experimental and astronomical timescales. Here the authors observe that polycyclic aromatic hydrocarbon fragments produced upon collision with He atoms at velocities typical of stellar winds and supernova shockwaves remain intact up to second timescales, thus may play an important role in interstellar chemistry. PAHs were first proposed as carriers of unidentified infrared bands at 3.3, 6.2, 7.8, 8.6, and 11.3 μm originating from interstellar dust in the 1980s8 and there are indications that they carry perhaps as much as 20% of the elemental carbon3. They are expected to form the basis for carbonaceous grains10, act as catalysts for the formation of molecular hydrogen16, and to be potential precursors for closed-caged fullerene molecules as large PAHs are broken down by energetic photons or particles18. However, the mechanisms leading to the formation of PAH molecules and fullerenes in interstellar and circumstellar environments are still not fully understood22.There is little doubt that large polycyclic aromatic hydrocarbon (PAH) molecules are ubiquitous in the interstellar medium (ISM)23, which help explain their abundance in harsh interstellar regions1. However, the processing of these molecules by UV radiation from young stars1 and by the impact of energetic particles from stellar winds and shockwaves25 can partially break them down and increase their reactivity. This is one of the key steps in models aiming to describe the formation of fullerenes from large PAHs, and energy deposition is also required for certain H2 production pathways involving PAHs16. When PAH molecules, or their fragments, are excited they can cool by dissociation, isomerization, or photon emission. Dissociation, a destructive process, will typically take place through the fragmentation channels with the lowest activation energies, i.e., through the loss of H, H2, or C2H2 units, all of which require about 5 eV for intact PAH precursors23. Other fragmentation pathways, such as the loss of a single C atom (or CHx unit), require significantly more energy and are statistically unlikely in cases where the internal energy is spread over all internal degrees of freedom of the molecule, for instance in photo-driven processes26. However, there are mechanisms that may favor otherwise unexpected dissociation processes. One such example is collisions with atoms or ions27. An energetic ion colliding with a PAH molecule—or any other molecule for that matter—will not just transfer kinetic energy in the collision, but also momentum that is localized to the point of impact. This momentum transfer can in turn lead to individual atoms being knocked out of the molecule in a pool-ball-like scattering process30.PAH molecules and ions are inherently stable systems with high ionization potentials and dissociation energies32. The most distinct fingerprint of these processes are fragments that have lost a single carbon atom, such as C59 and C23Hx fragments from C60 and coronene (C24H12) precursors, respectively30. In addition to these processes being observed with isolated molecules in the gas phase, knockout-driven fragmentation has also been shown to induce effective bond-forming reactions in cold, loosely bound clusters of such molecules that serve as laboratory analogs to small interstellar grains35. These grains may also be processed by energetic particles or radiation to form PAHs and fullerenes37 as part of the carbon cycle of the ISM38.Knockout-driven fragmentation has been identified in numerous experimental and theoretical studies of PAHs and fullerenes colliding with atoms and ions27, a common velocity range for stellar winds and shockwaves resulting from, e.g., supernovae25. It has therefore been proposed that knockout-driven reaction pathways may play an important role in the processing and evolution of complex molecules in the ISM , destroying the product of the initial knockout process. Such secondary fragmentation processes often lead to less reactive products26. Previous experimental studies have demonstrated that fragments produced by carbon knockout are indeed stable on microsecond timescales27, but until now it has not been possible to follow the fragments on radiative-cooling timescales (milliseconds or longer) in laboratory experiments.The quasielastic scattering processes that can lead to the knockout of atoms from a molecule dominate at impact velocities up to about 100 km/s40. The ultra-low residual-gas density in the storage ring, coupled with the ability to dump the beam of circulating ions after any number of revolutions, enables us to follow small numbers of ions over long timescales. This allows us to study the cooling dynamics and stability of the PAH fragments for seconds or longer. We find that a small part of the stored ion population dissociates spontaneously on timescales up to 10 ms due to their high internal energy, but that the majority of the fragments formed by the knockout of a single C atom do not undergo additional spontaneous fragmentation and remain stable for the full 1-s measurement. At this point, the remaining ions have spontaneously cooled enough through photon emission to remain intact in isolation indefinitely. This shows that damaged PAH species formed in energetic collisions with atoms or ions can survive in the ISM and thus can contribute to astrochemical reactions there.Here, we report on the storage of coronene cations, and fragments of these ions produced in collisions with He atoms at velocities of 72 km/s, in the cryogenically cooled electrostatic storage ring DESIREE∘ electrostatic deflector system and ion optics on the injection line of DESIREE is m/Δm ≈ 200, and thus not high enough to resolve products with different numbers of H atoms, but the peaks from fragments that have lost one or two C atoms, respectively, are clearly visible. The resolution is limited, in part, by the energy spread of the ions after the collisions. Since the measurements determine the kinetic energies of the ions, the mass scale is only a tentative scale to aid the reader. Also shown in the figure is a mass spectrum obtained from our classical MD simulations of the experimental conditions. A high-resolution mass spectrum  from the simulations shows that the knockout of carbon atoms is the dominant fragmentation channel . The two most common fragments are those that have lost CH and C, respectively. There are additionally some fragments resulting from the loss of two or more C atoms, together with different numbers of H atoms. Reducing the resolution of the simulated spectrum to match the experimental one results in the green spectrum in Fig. 23HA measured mass spectrum from colliding coronene cations with He gas at a center of mass energy of 105 eV is shown in Fig. 26. The next peak in the spectrum, that from the loss of two C atoms, is noticeably weaker in the theoretical spectrum than in the experiment. In the simulations, these fragments are solely the product of multiple knockouts by the single He projectile on sub-ps timescales. On the timescale of the measurement presented in Fig. 2Hx due to delayed fragmentation of intact, internally hot coronene ions contributes to the peak in the mass spectrum in addition to secondary C-loss from C23H3 K, which is enough for some of the molecules to spontaneously fragment on μs timescales. In addition, the molecules may be heated in collisions with He. In Fig. 41). For all of the different fragments, the excitation-energy distributions are broad. The distribution for the intact ion includes distant interactions with little to no energy transfer, skewing it to lower energies depending on the maximum impact parameter used in the simulations (this is why no median energy loss is given in this particular case). Fragments that have lost only H atoms or a single C atom (with any number of H atoms) have distributions with similar shapes and median excitation energies of 2.9 and 7.3 eV, respectively R is the measured rate of neutrals, and A and C are constants. The lines in Fig. 48. This typically occurs when a diverse ensemble of decaying units with many different time constants coexist. It has been shown that a tp behavior (with p ≈ −1) can result from an ensemble with a broad internal energy distribution, which in turn gives a population with many different decay rates at a given time, t42. From the data and fits in Fig. 24H23Hp ≈ −1.7) within the statistical uncertainties. The C22Hp = −2.2. For all three systems, this decay is only visible at short times, before the count rate reaches the level dominated by the constant detector background. From these measurements, we cannot draw any conclusions regarding the specific values of the exponents in Fig. 24H23HIn Fig. 2, or C2H2. But the measured rate can also be attenuated by radiative cooling, which is a non-destructive process that cannot be observed directly from neutral particle detection. Such a process can, however, lead to a deviation from a single power-law decay behavior by reducing the fragmentation rate at later times or, if the cooling rate by photon emission is higher than the fragmentation rate at all measured times, dictate the overall shape of the decay curve. The disappearance of the decay signal after 10 ms does not imply that the storage ring is empty of ions, but is instead due to the remaining population of ions being too cold to spontaneously decay (at least at a detectable rate). In this latter part of the decay curve the signal—i.e., the measured rate of neutrals—is too low to be distinguished from the detector background level. We can therefore not conclude whether any ions are left in the ring at long times from the decay curves shown in Fig. The only process that leads to neutral products as the molecular ions cools is fragmentation, which for PAH cations  most often takes place through the loss of H, H23Ht = 309 μs after ion production) and the longest measured time involved dumping the ion beam one second after production. During this time we can see that the number of C23H−2 s the decay abruptly stops as the population has cooled sufficiently such that the storage lifetime is limited by the experimental conditions, i.e., by collisions with the extremely dilute residual gas. Since the latter takes place at a rate proportional to the number of ions in the ring, it will result in an exponential decrease of the ion population with storage time. To identify the contributions from the two processes—dissociation of (some of the) stored ions and neutralization through collisions with the residual gas—we fit the following functional form to the data in Fig. N(t) is the number of ions remaining stored at time t, τ is the ion-beam storage lifetime, q is the exponent of the power-law decay at short times, and A and B are constants  ∝ −dN/dt. If we assume that the influence of residual gas collisions is negligible at short storage times, then the second half of Eq. (R(t) ∝ tq−1) at early times. From the fit in Fig. p = −1.76 ± 0.05 for the C23Hp can be accurately determined from a large number of data points we choose to use this value to set q = −0.76 when fitting Eq. (23Hτ = (70 ± 5) s from complementary measurements where the stored ions were dumped onto the detector after 100 s. The storage lifetime of ions in the DESIREE, which is typically on the order of minutes or longer49, depends on the ring parameters and vacuum conditions and does not reflect the inherent stability of the ions. This value was used as a constant when fitting the data in Fig. In order to investigate if Cmentclass2pt{minimmentclass2pt{minim23HB = 2.77 ± 0.03, which corresponds to the mean number of detected ions in the ring (per injection) that are too cold to spontaneously decay. Of the ions that are initially injected into the ring , close to 80% remain after the first 100 ms. We can thus conclude that the population of stored ions that give rise to the neutral products at early times, measured in Fig. The key finding from these measurements is that a significant fraction of the C23H23H23H26 do not dissociate, we see from the MD-simulations in Fig. 23H14H10) ions determined that approximately 10 eV of internal energy is required to induce fragmentation before the ions are stabilized by radiatively cooling51. If we use this higher value as an upper limit on the excitation energy C23H23H−16 cm2 for producing surviving C23HWe use our MD simulations to estimate the fraction of C24. In that work the authors model the processing of PAH-molecules in supernova shockwaves over a range of molecular sizes and shock velocities. They found that the dominating PAH-damaging mechanism at shock velocities below 75 km/s is impact by He or H24. These are the conditions that are reproduced in our experiments and for which we show that the resulting PAH fragments will stabilize on millisecond timescales. Depending on the gas column density and PAH size used in their model, Micelotta et al. found that each individual PAH molecule will typically collide with one to two He atoms with enough energy to remove a carbon atom before they are injected into the ISM24. At higher velocities (above 100 km/s) the authors identified that additional collisions with electrons will contribute to damaging the PAH molecules further. However, large PAH molecules (containing a few hundred C atoms) will be less sensitive to this mechanism, so even there the knockout mechanism is expected to be important.Here, we have shown that highly reactive fragments of PAH molecules damaged by the knockout of carbon atoms in collisions with energetic particles can remain stable on indefinitely long, i.e., astronomical, timescales in the gas phase. Extensive modeling work on the processing of PAH molecules by interstellar shocks has been performed by Micelotta, Jones, and Tielens24H2. With increasing PAH size the fraction of molecules that survive having a C atom removed from their structures will only increase. This is because the energy required to remove the atom will be independent of PAH size, as will the excitation energy remaining in the fragment, but the number of internal degrees of freedom in the molecule increases with size, reducing the rate of secondary fragmentation26. We can thus expect our findings to be directly applicable also for larger PAH species and related species such as fullerenes.The precursor system that we have studied here is the coronene cation, C26. For PAH molecules these defects introduce new structural features, e.g., carbon rings consisting of fewer or more than six atoms as shown in Fig. 41. In interstellar and circumstellar environments, the most probable species to react with PAH molecules damaged by carbon knockout would likely be hydrogen, carbon atoms or chains, or other hydrocarbon molecules (such as other PAH molecules and fragments)1. Molecular electronic structure calculations of intact and damaged PAH molecules have shown that the binding energy of an H atom to a large PAH molecule/cation  increases by a factor of four if a C atom is removed from the PAH molecule41. For other species, such as the aromatic phenyl radical (C6H5), the increase in binding energy for the damaged PAH is even greater41. The increased chemical reactivity of PAH molecules damaged by knockout processes could therefore play an important role in the growth of larger carbonaceous molecules and dust particles.The knockout process produces molecular fragments that are unlikely to be formed, e.g., in photo-fragmentation where the loss of a single C atom from a PAH molecule essentially is energetically disfavored52, but clearly, the knockout of individual C atoms by energetic particles, which is a highly efficient process, could very effectively lead to similar features. It is not yet clear what the relative importance of these different mechanisms are though, but further modeling or the identification of characteristic spectral features will be required to determine their potential contributions. The same goes for the increased chemical reactivity of damaged PAHs where the importance of such systems to chemical processes in the ISM remains to be seen. Nevertheless, it is clear that molecular fragments created by the knockout of atoms from complex molecules such as PAHs could indeed survive indefinitely in the gas phase, for example in the ISM.Another aspect is related to the formation of fullerene molecules in astronomical environments. One of the important steps in modeling fullerene formation from large PAH precursors is the conversion of hexagonal rings in the planar molecules to the pentagonal rings that are necessary to form a closed cage structure. This has been demonstrated to take place when photo-processing large PAH species53. DESIREE is outlined in Fig. 4 cm−3 (mostly consisting of H2), corresponding to a pressure of about 10−14 mbar40. In the present experiments, only a single storage ring was used, as shown in Fig. The experiments were performed using the Double ElectroStatic Ion Ring ExpEriment (DESIREE) at Stockholm University−2 mbar (chosen by optimizing for the knockout yield) was mounted on the beamline in order to produce coronene fragment ions from collision-induced dissociation. The 8 keV lab energy of the PAHs together with the He target resulted in collisions with energies in the C24H24 and stellar winds54. All fragments exiting the gas cell had close to the same velocity, so their kinetic energies scaled linearly with their masses. Products from the collision cell (intact ions or fragments) were mass/energy selected using a 90° electrostatic deflector system  just prior to the injection of the ions into the storage ring. This deflector system was also used to measure a fragment mass spectrum by scanning the deflection voltage together with the ion optics voltages along the beamline and measuring the ion current with a Faraday cup. When storing the ions, the settings on the deflectors and ion lenses in the storage ring were also adjusted accordingly from the setting optimized for the precursor beam depending on the mass-selected fragment. The first 10° deflector encountered by the ions that enter the ring was switched between injection and storage modes before an ion pulse made its first full revolution in the ring.Coronene powder (99.9% purity from Sigma Aldrich) was heated in an oven operating at about 230 °C, and the coronene vapor was fed into the plasma of an Electron Cyclotron Resonance (ECR) ion source. Ions extracted from the source were accelerated to 8 keV and chopped into pulses matching the length of the storage ring. Coronene cations were mass-selected using a 102° bending magnet  detectors located 1.2 m after the respective straight sections and independently counted as a function of the storage time. To probe the long time-stabilities of knockout fragment ions, the beam was dumped onto one of the detectors see Fig.  after a 56. The atomic parameters and the mixing terms used with this potential are from Stockett et al.57 Interactions between He and the atoms in coronene were described using the Ziegler–Biersack–Littmark (ZBL) potential58. The ZBL potential is a shielded Coulomb potential for describing the scattering of colliding atoms. The combination of these potentials has been successfully used in the past to simulate collisions between atoms and carbonaceous molecules like PAHs and fullerenes60. All of the simulations were performed using the LAMMPS software package61.We have performed classical MD simulations of coronene molecules colliding with He atoms in order to determine cross sections for different knockout channels and for estimating the internal energy distributions of the ions that we are storing in DESIREE. The interactions between C and H atoms were described using the Tersoff potential, a reactive many-body potential that allows bonds to be formed and broken between these types of atoms in a realistic way, taking into consideration the bond order of each atomxy-plane with its center-of-mass at the origin of a three-dimensional cartesian coordinate system. It is then randomly rotated around the origin at the beginning of each simulation to recreate the randomly oriented collisions in the experiment. A He atom projectile is initialized at z = 7 Å with the x- and y-coordinates individually randomized within a 12 Å × 12 Å box centered at the origin of the xy-plane. The simulation is started by firing the He atom in the negative z-direction with the same velocity as in the experiments (72 km/s), giving center-of-mass collision energy of 105 eV. Each simulation was followed for 5 × 10−13 s, long enough to follow prompt knockout processes, using a time step of 5 × 10−18 s. At the end of each simulation the positions and velocities of all atoms were recorded and analyzed to determine which bonds had been broken or formed, and the energetics of each fragment . A total of 105 simulations were performed, each with different initial parameters. A snapshot from a single simulation can be seen in Fig. The simulations were performed by first defining the structure of a coronene molecule, which is optimized to the potential-energy minimum in the Tersoff force field. In each simulation, the planar molecule is initially located in the 27. The potentials are defined for neutral systems, but for large PAH molecules with many delocalized electrons the differences in binding energies between neutral molecules and cations are small, so this effect does not significantly affect the outcome of the simulations when compared to the experiments.Using classical potentials, the simulations do not contain any description of the electron dynamics of the target or projectile. Likewise, the simulations do not contain any description of inelastic scattering resulting from interactions between the projectile and the electrons in the target molecule. At the velocities studied here, which are slow compared to the motion of bound electrons, this mechanism is relatively weak and plays only a small role in the overall energy transfer in the collisionsPeer Review FileDescription of Additional Supplementary FilesSupplementary Movie 1


The “mouthfeel” of food products is a key factor in our perception of food quality and in our appreciation of food products. Extensive research has been performed on what determines mouthfeel, and how it can be linked to laboratory measurements and eventually predicted. This was mainly done on the basis of simple models that do not accurately take the rheology of the food products into account. Here, we show that the subjectively perceived “thickness” of liquid foods, or the force needed to make the sample flow or deform in the mouth, can be directly related to their non-Newtonian rheology. Measuring the shear-thinning rheology and modeling the squeeze flow between the tongue and the palate in the oral cavity allows to predict how a panel perceives soup “thickness”. This is done for various liquid bouillons with viscosities ranging from that of water to low-viscous soups and for high-viscous xanthan gum solutions. Our findings show that our tongues, just like our eyes and ears, are logarithmic measuring instruments in agreement with the Weber-Fechner law that predicts a logarithmic relation between stimulus amplitude and perceived strength. Our results pave the way for more accurate prediction of mouthfeel characteristics of liquid food products. What drives the mouthfeel of ’thickness’? When is a soup too ’thick’? Here, authors measure the rheology of liquid soups and show their subjectively perceived ’thickness’ can be directly associated to their non-Newtonian rheology. Simple models, however, do not accurately describe the complex processes that liquid foods undergo in the mouth12, making it difficult to provide a deeper understanding of the relationship between the mouthfeel of liquid foods and its rheological properties. Importantly, the rheology of food products is rarely that of a Newtonian fluid with a viscosity independent of shear rate. This complicates the modeling of the complex  flow13 and lubrication15 in the oral cavity. Other complicating factors are interactions of the foods with saliva17, wetting of, and food deposition on the tongue and in the oral cavity10. The presence of particles also contributes19 to mouthfeel, with particle concentration, size, shape, and hardness20 affecting mouthfeel. When it comes to the descriptive sensory evaluation of a food texture, subjects can also have difficulties to properly assess a specific attribute because of the difficulty to disentangle a texture attribute from another21. For instance, “stickiness” refers to the sensation of food sticking to the palate and tongue during oral processing, but it is possible that subjects also assess this as “cohesiveness”, which is thought to be critical for the initiation of swallowing22.Mouthfeel of food products is a sensory determinant of product liking and repeat purchase. Prediction of mouthfeel sensory attributes  of food products is of paramount importance for the food industry but has proven to be difficult. It is usually attempted to link rheological parameters  of a stimulus is proportional to the logarithm of the physical stimulus intensity . While there is still a debate over Weber-Fechner vs Stevens’ laws, more recent work using a mathematical approach such as a Bayesian framework supports a logarithmic scaling of the response to a stimulus38 for auditory and number perception. However, when it comes to the question of mouthfeel, little is known about the psychophysical response of our tongue to the process of swallowing food. This is mainly because for liquid food products with a low non-Newtonian viscosity that depends on the stress and deformation rate, it is not clear under what conditions the food viscosity needs to be evaluated in order to relate it to the mouthfeel. For a long time it has been assumed that there is a single relevant shear rate in the oral cavity and many attempts have been made to correlate the perceived product thickness to the viscosity at this shear rate, also including rudimentary notions of shear thinning40.These complexities have prevented a complete understanding of even the simplest and yet one of the most important sensory attributes of liquid foods: the perceived “thickness”. The mouthfeel “thickness” is mostly linked to the viscosity of the food product, and sensory panels are relatively good in assessing the thickness of different liquid food substances25, Table Here we go beyond these simple models and develop a conceptually physical model describing the fluid mechanics of liquid foods in the oral cavity in order to establish relationships between rheological properties of shear-thinning, low viscous liquid foods and the mouthfeel perception of their thickness. To do so, we develop a physical model that captures the fluid mechanics, spreading, and rheology, as well as the biophysical aspects of eating and sensing. We test this model by investigating the rheology and sensory assessment by a test panel of 14 liquid bouillons varying in viscosity by three orders of magnitude, from 1 mPa s to ~1 Pa s  and as-bought commercial bouillon soups set 2) with varying viscosity. Salt (NaCl) combined with two different polymers (xanthan gum and starch) give the samples and the soups their consistency; the salt and polymer concentrations of the 14 bouillons and liquid samples investigated are given in Table  and as-b with varThe differences between the various liquid bouillons are mainly due to the thickeners used, xanthan gum, potato and corn starches, and their mixtures with xanthan gum; the shear stress and its variation as a function of the shear rate are highly dependent on their concentration.43. In other words, a person examines the liquid bouillon by pressing an amount against the palate with the tongue and moving it back and forth. In this way, the liquid is deformed. The extent to which the liquid product “resists” this deformation, in other words the shear stress, is perceived as the “thickness” of the product.To develop a model that can predict the mouthfeel sensory attribute thickness based on the rheological behavior of the liquid products in the mouth, it is necessary to understand how we “sense” liquids in our mouth. The basic assumption we make is that the thickness of the fluid is determined by holding some of it between the tongue and the palate Fig. ; the thi39 and Kokini et al.44, we model the oral cavity as two parallel flat plates spaced apart at a distance h from each other  than the palate . Due to the presence of the liquid in between, this field is called soft lubrication or elasto-hydrodynamic lubrication47 (EHL) and one can show that the deformability of the tongue makes that we can consider the palate/tongue system as two parallel rigid plates. The resulting deformation δ can be calculated from49:Rtongue is the radius of curvature of the tongue45, and FN is the normal force, also known as a lingual force. Substituting all numbers, we obtain deformation δ = 7 mm and a contact area of radius R = 2 cm. This means that the deformation is smaller that the characteristic size of the system Rtongue ~ 5 cm, in line with our assumption of two parallel plates without significant elasto-hydrodynamic effects.Based on the pioneering works of DeMartineher Fig. D). InterFN, while the lateral movement of the tongue at constant speed43, V, provides a shear rate h(t) changing in time. The earlier models do not properly incorporate the effect of the tongue speed on the evolution of the gap, but also neglect the complex rheological response of their materials . This calls for a reevaluation of the hydrodynamics of the swallowing process.Thus, in our model, the liquid is pressed between the plates with a normal force h(t) by adding the effect of the tongue velocity, resulting in what we refer to as dynamic squeezing (see Methods section). Note that we do not assume that there is a single characteristic shear rate, but instead consider in the model the full integrated response of all stresses and shear rates generated during consumption of the bouillon.We model the evolution of the gap σ at the surface of the tongue11: it is this stress that squeezes out the food product from the gap between the tongue and the palate. Based on the properties of the liquid foods that we consider here, the total viscous stress σ can be expressed in the rheological parameters κ and n:In our approach, we consider that the perceived thickness of the food product is proportional to the shear stress V while squeezing the liquid with a lingual force FN, giving for the variation of the gap:h0 is the initial gap between the tongue and the upper part of the oral cavity, R is the radius covered by the liquid product on the tongue . Different from previously derived models linking rheological properties to thickness perception44, the complexity of the flow profile is properly taken into account, based on the principle that shearing a shear thinning fluid makes it thinner, which implies that it is squeezed out faster. Note that if one considers the specific case of a Newtonian liquid (n = 1), the result becomes independent of the tongue velocity V, as expected51. To solve the equations, we use the fact that the speed of the tongue is higher than the velocity due to the squeezing. We discuss this approximation in more detail in Methods section and in Supplementary Note In the scenario considered here, the tongue is moving at a horizontal speed entclass1pt{minimaFrom Eq.  we can t9.Using the total shear stress assures that the effect of all shear rates is considered and, hence, that all dynamic viscosities are considered. This is a major improvement compared to earlier modeling approaches where correlations are made between thickness perception and a value of the viscosity that corresponds to a specific (but arbitrary) shear rateV0 ~ 4 mL that spreads over the tongue surface and is trapped between the palate and the tongue. The contact radius of the liquid cylinder between tongue and palate is estimated from the wet area made by licking a piece of paper , and found to be R = 2.5 cm liquid bouillon of volume χ of the two fits (see supplementary note 5) it is hard to decide which gives a better description. In fact, earlier observations by Demartine and Kokini et al. for paste-like foods favored a power law; however, their model clearly underestimates the shear stress, because of the simple approximation flow considered in their work (i.e. static squeezing). In addition in their work the input biophysical parameters were used as adjustable parameters because they were not known at the time  are in a good agreement with recent studies conducted on simple liquids looking at the deformation of model papillae11. Without adjustable parameters, the rheology of a fluid can predict its perceived thickness. Our study shows that the sensory attribute “thickness” is strongly influenced by the shear thinning, as all bouillons are rather strongly shear thinning, with viscosities that vary with shear rate over one to almost two orders of magnitude for a single liquid. The importance of these findings lies in relating the physical properties (i.e. rheology) of liquid food products to their mouthfeel. When coupled with the molecular and structural properties of the ingredients used, such as, for example, thickeners or biopolymers, such an approach may in the future enable us to design food in a way that it achieves the ideal mouthfeel. The results presented here focus on (semi) liquid materials; further experimental and theoretical work is needed towards predicting the mouthfeel of thicker or more complex products such as yield stress fluids with correct hydrodynamics.We conclude that the sensory attribute “thickness”, an important element in the mouthfeel of food products, is well predicted with our model which accounts for squeezing flow during liquid food consumption, and which suggests this subjective parameter is proportional to the logarithm of the shear stress on the tongue. This logarithmic relation between perception and stimulus, which has also been observed for other sensory organs such as our eyes and ears, agrees with the Weber-Fechner law. Despite its proven importance and impact on how we perceive things in our daily lives, such psychological law has never been reported before for the oral thickness perception. We are able to conclusively make this connection by considering a more realistic dynamic squeeze flow and by including the biophysical parameters that govern the swallowing process. Interestingly, the values of the calculated stress ∘C for further analysis. This temperature is specifically chosen in order to compensate the heat loss during the transfer of the liquid bouillons in the pre-heated soup bowl  and thus being served at a temperature very close to the mouth-temperature ~40 ∘C. For set 3, see25.We prepared sets 1 & 2 (see Table 56). Products were offered in a sequential monadic blind test using 3-digit codes. Two measurements were made for each sample, and the product order was randomized per replicate. To ensure minimal product variation (e.g. evaporation resulting in thicker products), panelists received the products in the same order. Breaks were provided in between the samples to allow for palate cleansing. Tap water and unsalted cream crackers were used as palate cleansers. The trained panel received 150 ml of product, served in preheated china soup bowls. Product was prepared directly before serving. The panel testing is done with a tablespoon of volume V0 ~ 4 mL. A modified spectrum method54 was used with a 16-point category scale (0–15). The panel was trained to score the intensity of all attributes according to salt solution references (see Table p < 0.05) on the attributes. Attribute list is reported in Supplementary Table 25. All samples are reported in Table A strict protocol was followed for the sensory assessment of the samples of sets 1 & 2 by an experienced, highly trained test panel of 11–14 members  allowed the control of the sample temperature during measurements. Rheology measurements were performed at 40 ∘C, which is the temperature assumed to be the relevant temperature inside the mouth before swallowing, and right after their preparation—as for the sensory analysis—since the samples can change in time due to the presence of starches and xanthan gum  and a 1The extensional properties of the solutions have been measured based on the principle of filament stretching experiment. A rheometer  was used as the building block of the device. A rheometer geometry plate with a diameter of 5 mm was used, the lower plate having the same diameter. The upper plate can be pulled vertically at a constant velocity until the capillary bridge breaks. A Peltier cell allows us to impose the temperature of the sample during the elongational process at a constant speed. The device is coupled to a high-speed camera  equipped with a microscope lens (Navitar). The profile of the neck diameter is automatically followed in time with a homemade Matlab routine by processing the raw image. More details and results are provided in Supplementary Fig. θ ~ 30∘, which is a likely result since they have similar biopolymer composition.To assure that the investigated liquids are initially trapped between the palate and the tongue, we measured the contact angle of a drop deposited on a piece of paper, with the help of a Drop Shape Analyzer from Krüss. For the measurements, we deposited 1 μL of each liquid samples. We find that the contact radius remains pinned and constant for all samples V in x-direction, and a rigid disk of radius R at the top, coming down with velocity Etongue ~ 2.5 − 10 KPa) when put in contact with the palate Epalate ~ 30 − 50 MPa. We assume the “thinning effect” causes the flow to be mainly in the x-direction, so uy = 0.The dynamic squeezing model used to predict the subjective thickness shown in Fig. 58 since the layer thickness h(t) is much smaller than the tongue radius R. Here, the z-derivatives are dominant, and pressure P is a function of x only:κ and n the rheology parameters defined as in Eq. (c1 is a constant determined by the second boundary condition. It is impossible to solve this analytically59. To solve the equations, we use the fact that the speed of the tongue is higher than the velocity due to the squeezing, The momentum equation for a shear-thinning liquid is found by using the classical lubrication equations in Eq. . Using tx is given by:V. This flux should be equal to the flux coming from the movement of the tongue, and the flux due to squeezing:h yields the final expression:The flux at any position Further information on research design is available in the Supplementary InformationPeer Review FileReporting Summary


P(N(t), t). Our results display that the model based in the Itô’s diffusion fits well to the results due to uncertainty in the official data and to the number of tests realized in populations of each country.The stochastic model for epidemic spreading of the novel coronavirus disease based on the data set supply by the public health agencies in countries as Brazil, United States and India is investigated. We perform a numerical analysis using the stochastic differential equation in Itô’s calculus for the estimating of novel cases daily, as well as analytical calculations solving the correspondent Fokker–Planck equation for the probability density distribution of novel cases,  On the other hand, the modelling of natural phenomena up to end of nineteenth century can be viewed as the study of deterministic solutions of differential equations7. At that time thought that whether all initial data could only be collected, one would be able to predict the future with certainly. We now know this is not so since that the concept of chaos has arisen, in which even quite simple differential equation systems have the rather property of giving rise to unpredictable behavior9. A historical example is the Brownian motion or the motion viewed of pollen grains in suspension in the fluid, discovered by the British botanic Robert Brown who observed firstly the motion in 18278. It was not quickly solved, and a satisfactory explanation did not come up to 1905 by Einstein. The same explanation was independently developed by Smoluchowski. The Einstein’s results were derived later by Langevin who writes in the form of the following differential equation a and b are known functions and x(t) and 9. In addition, we require a, and thus require that 10.The coronavirus disease (COVID-19) emerged in China in the end of 2019 year and since then it has spread reaching all countries in the globe. The quick spreading of the disease made the study of spreading of great importance for forecasting, control and prevention7. The logistic model may be modified so that unbounded growth does not occur. The simplest way to do this is to introduce a factor which will have the effect of making the variation rate of population growth to be z is large. Where the model is given in form 12. In a general way, if the size of the population at time t is t, we can have It is known that the logistic model of population growing can be used as model for some infectious diseases as the smallpox that are spread largely by individuals who can transmit the disease but who exhibit no overt symptoms16. The stochastic analysis was employed for statistical inference in a stochastic epidemic model for the Ebola virus in Democratic Republic of Congo in 1995, using the Monte Carlo method which is used to explore the distribution of the parameters in Ref.17. FFurthermore, a discrete-time stochastic epidemic model, with distribution binomial to study the transmission rate of the COVID-19 has been studied in Ref.18. Where the parameters of the model were estimated fitting to the novel data and using simulations, where one showed a trending of declining in the total number confirmed cases. In Ref. Cite Kaustuv, a stochastic model for health care impact of the epidemic of novel coronavirus in India has been studied in using Monte Carlo simulation, where the hospitalization, intensive care unit requirements and deaths were modeled and the impact of social measures distance and lockdown on checking of the epidemic was estimated. In Ref.19 was developed a stochastic approach described by the master equation and transition rates for the infection process. Here, we aim to use the stochastic nonlinear differential equation in Itô’s calculus for modeling of the dynamic of novel cases daily as well as the cumulative cases number since the beginning of pandemic until today. We use different statistical tests to give a future estimating of novel cases and behavior of the curve of the spreading of disease by solving the stochastic differential equation and using the probability density P, obtained analytically from solution of the Fokker–Planck equation. Due to large uncertainly in the official data about the real number of novel cases generated by the low number of tests realized in countries as Brazil, where the effect of randomness in the official data is modelled by the random term in the stochastic differential equation, being the use of this analysis hence, largely adequate to treat the spreading of coronavirus disease. As far as we know, there are none work that employs the modeling through nonlinear stochastic differential equations of Itô for the modelling of the spreading of the coronavirus. The plan of this paper is the following. In section "In this paper, we investigate the Itô’s stochastic diffusion model as possible model for the spreading of the COVID-19. The use of this type of analysis has been employed in the study of the behavior of time series for the price dynamics in Refs.t is N(t),we can have the growth of N as a function of t, subject to environmental random effects so that f(t) is a polynomial of n degree obtained from least squares fit to set of data. Furthermore, we have a deterministic term t of this term implying in a multiplicative white noise in Eq.  is another fit least squares to the set data within the range considered. The deterministic part g(t), can adjust to the reported novel cases in each day t using a n degree  of ItôN(t), of infected by COVID-19 registered in the Brazil as a function of time (days) registered in the period from 15th March, 2020 up to 12th August is displayed in Fig. g(t) given by t values reflects in an increase of the uncertainty in the data due to the low-number of test performed in the population. Consequently, for modeling of this behavior, we add the randomness term in the logistic model for growing of infectious disease as given by Eq. (g(t) is the adjustments to the set of data of COVID-19 of each country where the uncertainty after a time t large becomes larger generating so, an increase of distance between the real data. In Fig. N(t) for some countries as United States and India. The range of data considered is up to August 21. We obtain the time series of the model Eq.  as function of time t, t, with aim to calculate the analogous of the mean square displacement root T is the temperature of the fluid and In Fig. 11. Furthermore, at range of small t values, where the kurtosis is nearest to zero, we have that the distribution is nearest of a Gaussian distribution (We introduce the  the Eq. . Moreoveentclass1pt{minima the Eq. ) for theP, respectively8.Corresponding to the models Eqs.  and 2),,2), we c22. Here, we take the stochastic variable X be N, in the models Eqs. (X(t) is a stochastic process defined on probability space P a probability measure. That is, a function that to every set In a general way, for a nonlinear Fokker–Planck equation given bymentclass2pt{minimX, defined on B of P(X(t), t), given X a random variable on a probability space P is the probability measure of X. B of 9.The random variable W(t) is a Markovian process, presenting normal distribution, satisfying the conditions The Itô integral for X(t) using the Stratonovich integral is given byWe can write the Eq.  as7\docu the Eq. . The solX(t) is the solution of the following modified Itô equationx. Therefore, the Eq. (This implies that  the Eq.  in Itô ch(x) be a Borel-measurable function. Fix From the Feynman-Kac theorem, let t and x. Then g satisfies the partial differential equationx, where we assume that the stochastic process g(X(t), t), 9.Furthermore, we assume f(X(t)) using the Itô formula8From Eq. , we obtaTaking the average of both sides in the equation above, we obtainWe integrate by parts and discard surface terms to obtainK and a diffusion coefficient given as Thus, we have a complete equivalence between the diffusion process defined by a drift coefficient 8. However, it is different from Eq.  to make the connection with the Fokker–Planck equation being hence, different from prescription usually made.Even though in literature has been used the Stratonovich equation to make the connection with the Fokker–Planck equationN(t), f(N(t)). Using the Itô formulaWe can perform an analytical analysis solving the correspondent Fokker–Planck equation to the stochastic differential equation Eq. . We starIn following, usingP(x) is well behaved. We take the boundaries at infinity as 8. We use the initial condition Taking the Fourier transform of the above equation, we can guarantee the normalization of the probability density where For solving the Fokker–Planck equation time independent we make the power series expansion k is a separation constant. For We obtain the following recurrence relationsAdditionally, we haveP(x) in the formTherefore, we obtain To ensure the normalization of the probability density, For from Eq.  that \do the Eq. 36\documenth moments t. From the solution of the Fokker–Planck equation, we find an analytical expression for the mean half-width of the distribution as function of time given byWe find the In addition, we have , solution of the Fokker–Planck equation Eq.  in Brazil based in the nonlinear Itô’s diffusion model. Our results are compared with official data supplied by the Brazilian healthy agencies where due to large uncertainty in the results generated principally by the low number of tests made in the population and hence, to under reporting, generates a large uncertainly in the official results and consequently, the stochastic differential equation analysis becomes a more realistic model for growing of the total number of infected N(t). We can use an approach beyond white noise limit to try to better describe the expansion dynamics what can be done in a future work. The model reported here is based on Brazilian data from March 15, 2020 which shows an upward trend in the coming weeks.P and an analytical expression for variance of the distribution, where the standard deviation gives an estimating of novel cases daily and hence, we re-obtain all results obtained before, numerically by stochastic differential equation. In addition, we obtain a correspondence between stochastic differential equation and nonlinear Fokker–Planck equation obtained in the framework of the non-additive statistical mechanics different of the connection made in the literature24.Solving the Fokker–Planck equation, Eq. , we obta


Like all volumetric displays, OTDs lack the ability to show virtual images. However, in this paper we show that it is possible to instead  This display operates by first confining a particle in a photophoretic trap which then drags the particle through every active image point in a freespace volume. In the current prototype, this volume is typically 1 cm3. As the particle is dragged through space, it is illuminated by visible lasers to form an image by persistence of vision. The volume must be redrawn more than ten times a second to be above the flicker rate of the eye. The most immediate current challenges to bring the display into the realm of practicality are (1) to scale the display volume from 1 cm3 to greater than 100 cm3 and (2) to employ parallel traps2 to address the fundamental incapacity of freespace volumetric displays to create virtual images. Our efforts to address the former challenge will be reported in another publication. Instead, this work focuses on the latter challenge, the challenge of simulating virtual images with freespace volumetric hardware.In 2018, the authors presented a new platform for volumetric display using photophoretic trapping3. The physicality of these volumetric points gives them perfect accommodative cues . This definition requires that volumetric images be composed of real image points that must exist only in a finite drawing volume. So, to display an optically correct volumetric image of the moon seen through a window would require the OTD display to be scaled to astronomical proportions. This situation is not unlike that of a movie set or theatrical stage, where props and players must occupy a fixed space even when trying to capture a scene meant to occur outdoors or in outer space. In the theater, this limitation is mitigated by including a flat backdrop that contains pictorial 3D cues such as a road winding to a point (perspective cues) or mountains eclipsing one another (occlusion cues) as they fade (atmospheric cues) into the distance in order to create the sense of enlarged space. In a modern theater production, using projections for backdrops, motion can also be used to simulate parallax. This is effective because the background depicts sites at distances where the focus cues like accommodation and vergence would not be dominant. This approach could also be used to simulate virtual images. Volumetric images share these challenges and could share these solutions. With purely real volumetric image points, freespace volumetric displays will be forever confined to the drawing volume. What is needed is a ‘backdrop’ for volumetric displays.Also, photophoretic optical trap displays have a fluid drawing volume, which means that it is theoretically possible to create images that are much larger than the display itself. However, there are practical limits to how large a volumetric image may become. Volumetric images are defined as having image points co-located with physical point scatterers2 points out the limitation that all image points must lie along a line extending from the observer through the display volume. The points that the user perceives in the back plane are no longer volumetric because they no longer coincide with physical scatters, so they lose the attribute of perfect accommodation6, but they gain the ability to dramatically increase the perceived size of the image volume. Using a perspective projection, an OTD can simultaneously generate both real volumetric image points for the foreground and simulated, non-volumetric image points for the background, greatly expanding the usefulness of the OTD platform.In this work, we apply and extend the backdrop analogy to simulate virtual images in a photophoretic optical trap display (OTD). OTDs can draw flat and 3D structures in air Fig. b,c. It i7. However, even in this small volume, it is possible to create a simple proof-of-concept experiment for simulating virtual images. We also feel compelled to do so as quickly as possible because the inability to create virtual images is so fundamentally limiting to the entire freespace family. Therefore, the result of even this small-scale experiment literally and dramatically expands the application space for all freespace volumetric displays capable of angular scatter control.The small display volume makes the current prototype of limited utility, and we will have to wait for larger parallel prototypes to test many display-level characteristics9. This paper uses cellulose particles estimated at 10 μm1. When the trap is moved, the particle is dragged along with it. The particle is moved through all of the image points in succession. When the particle reaches an image point, it is illuminated with a combination of red, green, and blue light. The particle moves through every point in the image several times a second creating an image by persistence of vision , the image point to be displayed X = , and the plane on which to display P. Finding the intersection of the line EX with the plane P gives the pixel coordinate of the point X. The perspective projection can be defined by the following matrix relationship for a plane P perpendicular to the line EO where O is the origin:One of the most general forms of perspective is ray tracing where the observer or camera is considered as a single point E = , we drew a flat 2D) OTD image of the moon at the back face of our drawing volume. This plane, in turn, sat at the front face of a 3D printed miniature of a house  can be used to eliminate stereoscopic vergence and stereoscopic conflict.The monocular system employed in this experiment is convenient for proving the utility of perspective projection. However, it obscures somewhat the fact that focus cues, such as accommodation and vergence, and stereoscopic cues will be in conflict with other visual cues for the proposed “backdrop” voxels . The radius of swing was 100 mm to the front face of the camera lens. The house had dimensions of 7.7 × 10.6 × 7.4 mm. The moon had a diameter of 0.5 mm (which varied during the experiment). The moon was updated at 12 frames/s. The draw volume measured 0.5 mm in y and 9.2 mm in x (z volume was not used).The OTD drawing function was modified by perspective projection in synchronization with movement of the camera arm. The speed of panning was approximately 0.0194 m/s. Camera and lens used were a Canon EOS 6D and Canon MP-E 65 mm f/2.8 1–5× macro lens, respectively. The camera was focused at the chimney of the house (approximately z = 0 mm). The moon is not modified as the camera rotates, providing a control image. In Fig. z = 0, but the moon is shifted laterally as the camera rotates to give parallax consistent with an object perceived at z = 8 mm. In Fig. spc is one component of the simulation pixel coordinate vector and epc is the corresponding component of the experimental pixel coordinate vector. Taking the Euclidean Average of the two error components gives an error of 5.88% in the image. In this experiment we increased the display space by 80% to 1.8 cm in one dimension compared to the physical volume of 1 cm3 of the display.In Fig. The modified parallax does appear to create images perceived behind the drawing volume. Our calculated error supports the use of this method. The modified parallax, after accounting for bias, shows good agreement with simulation. This shows the potential effectiveness of increasing the display space of the volumetric display beyond the physical boundaries of the display. The increase of display volume by 80% in one dimension demonstrated here can be extrapolated to infinity, given an immersive display where the viewer is always looking through the display volume.9 and we have outlined two possible methods for exerting control over this directional scatter in the future12 that would allow for each eye of the user to receive a different perspective based on their respective spatial locations. With the possible future addition of directional output control, the method proposed here would become more effective without any additional changes needed. The second limitation is that this method requires the viewer to be tracked ; this is a significant encumbrance as normal OTD real images require no knowledge of the user’s position and still provide almost 4π steradians of view angle. However, we can say that once directional scatter has been achieved, tracking of the viewer could be omitted in at least two dimensions . The angular outputs of the display having image points corresponding to the perspective from that position updated regardless of viewer presence. The third dimension of the viewer position, the distance of the viewer from the display, would still be needed for ideal perspective reconstruction as the perspective projection is based on a 3D observation point. Further pursuit of directional scattering control is thus capable of solving one major shortcoming of OTD technology at this time, reducing the complexity of the method presented here, and extending the usefulness of the method presented here to include independent virtual images for several viewers at once. The final limitation is that of mismatch between the accommodative cue, which leads the user to focus at the projection plane, and the parallax cue, which leads the viewer to focus at the perceived point. This stereopsis/accommodation mismatch is common in other systems14 sometimes causing adverse side effects to users16. To mitigate it, we must place the perspective projection plane at a distance where parallax is more dominant than accommodation. This requirement is in harmony with the theatrical backdrop approach that we have proposed in this paper, especially given the relatively rapid drop-off of accommodation dominance with image distance17.Limitations of this approach include (1) a lack of binocular disparity, (2) requirement of motion tracking of the viewer’s eye position, and (3) mismatch of accommodation/vergence and other visual cues. To the first limitation, this experiment was a monocular test. To be effective for normal-sighted human viewers, our approach must eventually be modified to also provide accurate binocular parallax. For binocular parallax to function, the OTD must be capable of controllable anisotropic scatter. To date, we have demonstrated anisotropic scatter3. Unlike OTDs, holograms are extremely computationally intensive and their computational complexity scales rapidly with display size. The complexity also scales rapidly with point spread function. Neither is true for OTD displays. Consider a background of stars: regardless of the number of stars, a holographic display would require terabytes per second of data to provide the diffractive focusing power to render sharp star-like points, and the parallax and focus cues would be wasted given the extreme distance of the virtual points. By comparison, OTDs would only require a bandwidth proportional to the number of visible stars (1.8 Mb/s to represent the approximately 5000 visible stars)We would argue that, these limitations notwithstanding, simulating virtual images with OTD would be preferable to the use of a hybrid OTD/holography system, which has been proposedFurthermore, we would argue that both volume images and perspective projection surfaces are best used in harmony and not to the exclusion of one or the other. For example, it would not be prudent to collapse the entirety of a scene, foreground and background, to a single plane and thereby lose the rare advantages of perfect accommodation and close-up viewing.We have demonstrated a display-level application for OTDs for the first time and have shown agreement with simulation to produce an effect similar to virtual images in optical trap displays. This result leads us to contemplate the possibility of immersive OTD environments that not only include real images capable of wrapping around physical objects (or the user themselves), but that also provide simulated virtual windows into expansive exterior spaces. See Fig. Supplementary Information 1.Supplementary Video 1.Supplementary Video 2.Supplementary Video 3.Supplementary Video 4.Supplementary Video 5.


The extent to which rates of homicide by people with or without schizophrenia vary over time has theoretical and practical implications in understanding homicide by people with mental illness.The aim was to report on the rates of homicide by people diagnosed with schizophrenia over time in a region in which there were dramatic changes in the overall rates of homicide.An examination of homicide by people diagnosed with schizophrenia in the course of judicial psychiatric examination, and the rate of other homicide in the Chuvash Republic of the Russian Federation between 1981 and 2020 was undertaken.r = 0.503, d.f. = 38, P = 0.001).During the 40 years of the study a total of 5741 people faced legal proceedings for a homicide offence, of whom 179 (3.1%) were diagnosed with schizophrenia. During the study period the average annual total homicide rate rose from about 9 per 100 000 in the 1980s, peaked at 17 per 100 000 in the 1990s before falling to 13 per 100 000 in the 2000s and 6 per 100 000 in the 2010s. Rates of homicide by people with schizophrenia also rose and fell over this period and were significantly associated with the rates of other homicide (The rise and fall in rates of homicide by people diagnosed with schizophrenia in parallel to total homicide suggests that homicidal behaviour might not be intrinsic to the clinical manifestations of the illness, and might instead reflect a heightened vulnerability to social factors that are associated with homicide by people without schizophrenia. This study contradicts earlier views about the stability and causes of rates of homicide by people with schizophrenia and suggests that other explanations, including the possibility that rather than being intrinsic to the psychopathology of schizophrenia, people with schizophrenia might be more vulnerable to the same risk factors that determine homicide rates in the general community.2 The implications of this finding include that homicide by people with schizophrenia might best be prevented by limiting the exposure to factors known to be associated with violence in the community, such as access to firearms and limiting alcohol and other drug use, as much as by treatment and hospital admission. One limitation to the 2009 meta-analysis2 was that it examined aggregated homicide rates over the duration of the included studies and did not examine changes in rates or in the relationships between the rates of homicide by people with and without schizophrenia over time.There is a well-recognised but poorly understood association between schizophrenia and homicide offending. Meta-analyses of available studies show a fourfold increased risk of homicide offending among men with schizophrenia and an eightfold increased risk among women compared with rates of homicide offending by people without schizophrenia,6 Another pattern was observed in New Zealand between 1970 and 2000 where the total homicide rate rose annually between 1970 and 1990 before falling in the last decade of the study, whereas the proportion of homicide by people defined as ‘mentally abnormal’  fell throughout the study, initially because homicide by people who were not mentally ill was rising and later because mental illness verdicts were falling faster than other court outcomes.7Several studies have examined the trends in homicide committed by a broader class of people considered to have a mental disorder. In England and Wales, rates of homicide by people with mental illness  rose steeply with rates of other homicide after World War Two, before falling just as steeply, while the rate of other homicides continued to rise.8 In Iceland homicide by people with mental illness  rose more quickly than rapidly rising rates of other intentional homicides in the period 1970 to 1984.9 More recently, a study from Ontario, Canada found a non-significant fall in homicide offending by people who were mentally ill during a period when the rate of other homicide fell more.10 To date the only study that focused on changes in rates of homicide by people with schizophrenia found a rise in the absolute number of homicide offenders with schizophrenia during a period in which other homicide rates fell in England and Wales.11 The major limitation in interpreting this group of studies as an indicator of the association between homicide offending by people who are mentally ill compared with those who are not mentally ill is the differences between studies in the way mental illness is diagnosed and defined in law between jurisdictions and over time.12In Denmark, homicide by people with psychosis (defined using ICD-8 criteria) were reported to be stable during a period of rising homicide rates between 1956 and 1970, but then increased in parallel with total homicide rates between 1970 and the mid-1980s.15 Previously we found that almost half of all homicide offenders had at least one mental disorder, including alcohol dependence experienced by 16% of offenders and schizophrenia experienced by over 4% of offenders.15 In this paper we report on the trends in rates of homicide by people with schizophrenia and without schizophrenia (other homicides) during a period of 40 years when the rate of total homicide rose and fell dramatically. We hypothesised that if rates of homicide by people with schizophrenia were fixed over time that they would not be correlated with the rates of homicide by people without schizophrenia and that the proportion of homicides by people with schizophrenia would fall while other homicide rates were rising and would rise when other homicide rates were falling. In other words, the rate of homicide by people with schizophrenia did not change when the overall rate of homicide changed .We have previously reported on the characteristics of offenders with schizophrenia and other mental disorders from The Chuvash Republic of the Russian Federation, a region with both a high total homicide rate and a high rate of homicide by people with schizophrenia.2, centred in the city of Cheboksary on the Volga River about 650 km east of Moscow. A third of the population live in Cheboksary and 40% are considered to be rural. During the period of the study, and as in the rest of Russia, people in the Chuvash Republic who are charged with a homicide offence are referred for psychiatric evaluation under Section 21 of the Russian Penal Code.16 Referral for judicial psychiatric examination by a panel of three psychiatrists is at the discretion of the investigators and the court. Some accused are not referred because there are no doubts about their sanity, and some individuals who are accused refused to be examined. However, most accused are examined, and the sample includes the majority of offenders referred for a formal evaluation, including all offenders suspected of having a psychiatric disorder. Data was also available about homicides that were immediately followed by the suicide of offenders known to mental health services.The study examined a data-set of all homicides in Chuvash Republic of the Russian Federation between January 1981 and December 2020. The Chuvash Republic has a population of over 1.2 million living in an area of around 18 000 km17Demographic, clinical and criminological data were extracted from the case files, which included the judicial psychiatric report, hospital records, out-patient notes and legal documents. About one-third of the offenders had been personally examined by either A.G. or A.T. The demographic data extracted were age, gender, occupation, marital status and years of education and the clinical data included psychiatric diagnosis, age at onset, previous treatment, pattern of symptoms, history of brain injury, and substance misuse, and the criminological data included previous convictions, method of homicide and judicial outcome. The psychiatric diagnoses were made using the version of the ICD that was current at the time of the judicial examinations. For the purpose of this study the psychiatric diagnoses in each individual was made by retrospective chart review using ICD-10 criteria.Non-age-adjusted homicide rates were calculated using the number of annual homicide incidents and the annual population of Chuvashia, as measured by the census or imputed using linear assumptions in the intervening years. The association between rates of homicide by people with schizophrenia and other homicide and the association between the rates of other homicide and the proportion of all homicides committed by people with schizophrenia were examined with a Pearson correlation coefficient using untransformed homicide rates.10 (1 + homicide rate per million population per annum) so as to avoid the scale minimisation effects that can arise because of the low base rate of homicide offending by people with mental illness.18 Possible differences in the characteristics of male and female homicide offenders with schizophrenia were examined using t-test and chi-square (χ2). When one or several cells contained five individuals or less, a Fisher's Exact test was used. All tests were performed in a two-tailed form. The data were analysed using SPSS Version 27 .The changes in homicide rates by people with and without schizophrenia and changes in the proportion of homicides  committed by people with schizophrenia over time were examined using Kendall's Tau rank correlation coefficient over the whole period of the study and over two equal periods (1981–2000 and 2001–2020) in which the homicide rates were, respectively, rising and falling. In order to illustrate the association between homicide in schizophrenia and other homicide, rates were displayed as LogApproval for the research was obtained from the Ethics Committee of the Medical Faculty of Chuvash State University, and the Chuvash Association of Psychiatrists, Narcologists, Psychotherapists and Medical Psychologists (a branch of the Russian Society of Psychiatrists). As the study only uses statistical information, no oral or written consent was taken from the individuals of the study.18In the 40 years from the beginning of 1981 and the end of 2020, 5741 people faced legal proceedings for a homicide offence, and 3410 homicide offenders were referred for psychiatric evaluation. Over this period there were more homicide victims than offenders because the police did not solve every case, some offenders had multiple victims, and some offenders died by suicide or died prior to being examined. A small number of offenders from other parts of Russia were examined elsewhere. The homicide clearance rates, defined as an offender being identified by the police, were more than 90% in the period of the study. Of those subjected to judicial examination, 46.7% were found to have at least one mental disorder, and 7.7% were found to be not responsible for their actions because of mental disorder according to the Russian Criminal Code.During the study period 179 homicide offenders were identified as having schizophrenia, including 14 recidivist offenders who committed both homicides between 1981 and 2020 . The pop1981–1990: average 8.9 per 100 000 person-years;1991–2000: average 16.4 per 100 000 person-years;2001–2010: average 12.5 per 100 000 person-years;2011–2020: average 6.1 per 100 000 person-years.During the four decades of the study the rates of homicide by people without schizophrenia rose and fell:1981–1990: average 0.28 per 100 000 person-years;1991–2000: average 0.41 per 100 000 person-years;2001–2010: average 0.47 per 100 000 person-years;2011–2020: average 0.21 per 100 000 person-years.Rates of homicide by people with schizophrenia also rose and fell in the same periods:r = 0.503, r2 = 0.253, F = 12.9, d.f. = 38, P = 0.001) .The annual rate of homicide by people with schizophrenia was strongly correlated with the rate of other homicide (Pearson's = 0.001) . There wP = 0.04) as did those without schizophrenia (τ = .67 P = <0.0001). In the period 2001–2020 rates of homicide by people with schizophrenia fell significantly (τ = −0.35 P = 0.03) as did those by people without schizophrenia (τ = −0.86 P < 0.0001) (In the period 1981–2000 rates of homicide by people with schizophrenia rose significantly (tau (τ) = 0.33  0.0001) .P = 0.18), during the period 1981–2000 when rates of homicide were rising (τ = −0.05 P = 0.75) or in the period 2001–2020 when rates of homicide were falling (τ = 0.06 P = 0.7)The proportion of homicides by people with schizophrenia did not change over the 40 years of the study  by women with schizophrenia. Adolescents  with schizophrenia committed 11 (6.2%) homicide offences . There wThis study examines the rates of homicide by people with schizophrenia over a period of 40 years in a region that had a dramatic rise and equally dramatic decline in the rate of homicide. The results confirm the well-recognised over-representation of people with schizophrenia among homicide offenders, but also suggest that the rate of homicide by those with schizophrenia is correlated with changes in the rate of other homicide and that the proportion of homicides committed by people with schizophrenia might not fall when other homicide rates are higher and might not rise as other homicides become less common.Although the reasons for the five- to tenfold increase in homicide risk associated with schizophrenia remains uncertain, the findings of this study suggest that the increased rate is at least partly because of a greater exposure or greater sensitivity of people with schizophrenia to more general risk factors for homicide, including factors such as substance use and social disadvantage, rather than an intrinsic or inevitable propensity to lethal violence associated with the illness.r2 statistic indicated that about one-quarter of the variance in the rate of homicide associated with schizophrenia was explained by the changes in the overall rate of homicide. This can be compared with the results of a 2009 meta-analysis that found that three-quarters of the variance in rates of homicide in schizophrenia between jurisdictions was explained by the rates of other homicide.2 Hence both cross-sectional and longitudinal studies suggest that homicide rates by people with schizophrenia are not fixed over time and between regions and that social factors that influence homicide by people without schizophrenia, for example, alcohol and other drug use, levels of violence in the community, and the efficiency of the police in preventing crimes influence the likelihood of a person with schizophrenia committing a homicide offence. However, the moderate R2 value in the association between rates of homicide in people with schizophrenia and other homicide does suggest that there are other factors that influence the rate homicide in schizophrenia that operate independently to those associated with total homicide. These might include the availability and quality of psychiatric care, variation in the prevalence of schizophrenia, and the specific impact of substance use on people with schizophrenia.In this study the 19The decline in the total rate of homicide in Chuvashia in the past 20 years is striking, and may be because of greater economic prosperity and the re-establishment of social order after the turbulent glasnost years. The decline may also be related to the ageing of the population, as many people of working age have moved elsewhere, and a rise in the proportion of women in the population. There was an increase in the number of people known to have schizophrenia in Chuvashia during the years of the study that might be as a result of an increase in the identification and reporting of patients with schizophrenia by a larger mental health workforce. Hence, it is also possible that an increase in availability of treatment contributed to the decline in homicide offences by people with schizophrenia, as suggested by the near elimination of recidivist homicide offences by people with schizophrenia in the past decade and the over-representation of recidivists in rural areas where services are less available. During the period of the study there were no changes to the laws governing psychiatric treatment, which has been proposed as a possible reason for changes in rates of homicide by people who are mentally ill in the USA, along with the quality of services.20 during a period in which the national per capita alcohol consumption fell by more than 40%.21 About half of the offenders with schizophrenia  were reported to be intoxicated with alcohol at the time of the offence, a pattern that had not changed in the decade since an earlier report.14A decline in the consumption of alcohol in Chuvashia might also have been important. During the period of the study alcohol consumption in Chuvashia fell from being above the Russian national average to well below the national averageStrengths of this study are the complete nature of the sample and related statistical information, consistency in the way the psychiatric examinations were recorded and an absence of changes to relevant sections of the legal code. Limitations of the study include that some homicide offenders did not receive a judicial examination, not all homicides were solved by the police, and the absence of information about the mental state of some of the homicide offenders who died by suicide before they could be examined.A more reliable estimate of the extent to which rates of homicide by people with schizophrenia vary from other homicide rates might be obtained by meta-analysis once further longitudinal studies have been conducted in other jurisdictions. The extent of this correlation has implications for our understanding of the reasons for homicide by people with schizophrenia, and the degree to which systems of treatment for psychosis, and more general measures to reduce violence and homicide in the community might play a role in the prevention of homicide by people with emerging and established mental illness.


This work investigated reasons and factors that cause the failure due to mechanical wear (Erosion) for the inside surface of elbows and pipes used in cement transportation which manufactures from low carbon steel and finds out a method for reducing this failure. The technique of Nano-coating layers is used to coat the surface of samples with layers of nanoparticles of tungsten carbides of different thicknesses of . The test was done for these samples by placing them inside the elbow under the same operating conditions, pin on disc test. The results of the test under the same operation condition showed a decrease in erosion rate by 71% for the sample coated with 50 μm of layer, while the results of the pin on disc test showed a decrease in erosion rate by 97% for the thickness of 50 μm as this test is done under ideal testing conditions. The decrease in wear rate for elbow and pipes will increase their life work two times at least and that reduces the cost of maintenance by about 75%. The numerical simulation was also implemented to simulate the erosion profile inside the elbow, and the agreement with experimental results was 90%. Many fine powders, such as cement, which exhibit very low de-aeration rates are suitable for dense-phase transport. Observation of the flow patterns, in horizontal pipes, when these powders are transported in dense phase reveals a stratified flow. A high concentration layer of fluidized material occupies the lower portion of the pipe. In the upper portion of the pipe, particles are suspended in the transport gas. The fluidized layer generated at the top and bottom of the pipes makes the particles suspended in the transport air2. Pipelines are important for transporting gases, liquids, and solid materials for long distances from their main source to warehouses or silos. The cement industry is one of the important industries in the world, as it is involved in a lot of applications in building and construction3. The erosion-corrosion rate investigated 30°, 60°, and 90° carbon steel elbows in a multiphase flow containing sand particles. Qualitative techniques such as multilayer paint modeling and microscopic surface imaging were used to scrutinize the flow accelerated erosion-corrosion mechanism. The results show the erosion rate for the elbow of 90° higher than in low angle elbows (30° and 60°), also the high erosion rate occurs in the upper half of the elbow4. The transport of these conditions at high pressures up to 7 bar where the flow is turbulent and the particles of cement distributed directly across the cross-section of the pipe because of the high level of turbulence. Solid particles have a high impact on metal surfaces; causing damage to the surface and remove parts from it. This phenomenon is called mechanical wear (Erosion). Erosion always occurs when the solid particles directly hit a solid surface. The mechanical erosion (wear) inside pipes plays a crucial role in the design and operation of the transporting system5. Investigations of wear are done for areas of interest of different industries. Earlier efforts in the field of corrosion research began in 1960 in the industrial countries; this shows the complications of the wear phenomenon6. The mild steel used to study the wear occurred in pipes, where a mixture of compressed air and abrasive solid particles was made. Their results indicated that the decrease in the bend curve leads to a decrease in wear7. The erosion in mud transport tubes using ANSYS software was studied numerically; the results showed that the erosion rate increases eight times8. The imperative used to simulate Nanoindentation by powerful FEM software to extract plenty of mechanical properties like hardness, elastic modulus, endurance loads and various parameters like optimal thickness and optimal critical load, stress distribution and contact pressure between substrate and layer can be obtained through load–displacement curve A9. The wear resistance and scratch were investigated to St-37 mild steel after coating it with a nanoparticle of nickel-boron. The method used is electrical deposits and heat treatment with a period of (1 h) at a temperature of (400 °C); the results showed a decrease in the wear rate of about 50%, while hardness increased by about 28%10. An experimental study on pipelines was conducted to find out the reasons that lead to erosion by using a sandblasting device for different values of flow speed (20–80 m/s). The results showed that at a high speed and for a long time, the erosion was an abrasive mechanism, whereas at a low speed the erosion was due to plastic deformation11. A numerical study on mild steel was conducted to find the wear. A steel pipe of 1.5 m in length and 50–250 mm diameter is used in their study. The sand is mixed with water to form slurry and to control the speed from within the range of 2–8 m/s. The results showed that the greater the bending angle, the greater the wear. The largest wear ratio was at an angle of 30°–60° degrees12.The Nano coating layer of Tungsten Carbide is used to coat the surface of high chromium cast iron by using a thermal spray process and studied the improvement in the wear resistance and microhardness. The result showed that the microhardness increased by a ratio of 34% and wear resistance increased by 79% due to the good wear resistance properties for Nanopowder of tungsten carbide and good adhesion properties with steel surfaces at high temperature in the thermal spray process14. It appears from the aforementioned literature review and to the best of the Authors’ knowledge, the study of the mechanical wear caused by cement particles passing through a 90° degree pipe elbow with a unique method to reducing this mechanical wear by coating the internal surface of the elbow with layers of nanoparticles of tungsten carbides (WC) has not been studied extensively yet. The objective of the present work is to find the technique to reduce the wear rate of elbow and pipes to increasing their work life and reducing the maintenance cost and reducing the production stopped. The experimental and numerical study would be a considerable contribution to the open literature and conducted the improvement in wear resistance for coated inner the surface of elbows and pipes that used in pneumatic conveyors for cement transportation.The term slurry defines as a "Non-homogenous mixture of solid with liquid, gas or air". The flow of cement and high-speed air mixture may cause mechanical wear for the internal surface of the pipes and elbows and that leads to big losses in money and time. The particle sizes of a solid vary from few microns to few millimeters. The effect of a high particle concentration is influenced by different factors such as density amount, size, mass part, particles of solid and carrier density; and mortar can be classified as an unstable factor depending on the size of the solid particle. The ability of solid particles to stabilize the carrier air is lowThe metals of low carbon steel are used in the manufacturing of pipes and elbows in most of the cement plants. The chemical composition of low carbon steel listed in Table3, melting point of 2870 °C, boiling point of 6000 °C. Figure The Nano particles of tungsten carbide (WC) has Nano-size of (55 nm) used as a coating material to coat the low carbon steel samples. This nanoparticles has true density of 15,500 kg/m14, shown in Fig. –2 g/s).High Velocity Oxy-Fuel (HVOF) is used for metallic and non-metallic (ceramic) materials of the surface layer to form a coating layer for a semi-molten or molten state. Fuel oxygen and acetylene are both fed into the chamber by a gas flow rate ratio of 3:1 for oxygen and acetylene applied with a pressure of 10 bars for oxygen and 5 bars for acetylene. Both gases and nanoparticles are controlled by special valvesThe samples used to carry out the test inside the elbow prepared from low carbon steel with dimensions of 70*30*5 mm and inner surface roughness of 0.9 mm, and the location of samples the inside elbow is set to ensuring that they exposed to real harsh operating conditions and the flow is easy to run and pass through the forms to ensuring contact at every point on the surface of each sample. Furthermore, holes drilled into the sample to make it easier to install inside the elbow with the best types of fastening, shown in Fig. Four holes of 8.5 mm in diameter drilled in the elbow to fixing the samples.Based on the pre-simulation results, a 10-degree cut angle was made with the x-axes at the front side of each sample .The steps of fixing the samples inside the elbow are as following:The site chosen in the inner surface of the elbow, which is expected to subject to the most severe conditions by cement particles and flow velocity ; and the samples placed on the inner circumference of the elbow using steel bolts, shown in Fig. 16. The surface of samples smoothed in the laboratories of the faculty of Engineering at the University of Kufa/ IRAQ. The pin rotation diameter is chosen according to the rotating speed  and the applied load on the pin is 20 N according to the standard specification, the abrasive wear occurs by mechanism of (deformation and cutting) for the tested samples surface by the pin from hard metal and abrasive particles will be removed continuously due to the high rotation speed as shown in Figs. The samples preparing for wear test (pin on disc) from low carbon steel of diameter (40 mm) and thickness of (10 mm) according to the standard specification of (ASTM G-99-17)eed 400, 00 r.p.m.17:The first step in predicting erosion is the flow field simulation. Fluid flow modeled by time-averaged Navier–Stokes equations to calculate the flow fields presented. RANS method was used to model the turbulent flow. Airflow assumed incompressible, Newtonian fluid with constant fluid properties, three-dimensional, unsteady, and in turbulence mode. Hence, the continuity, momentum, and turbulence equations for an incompressible fluid, three-dimensional, and time-dependent areMomentum equations:where: U is the time-averaged speed and Continuity equation:18:In order to find the turbulent eddy viscosity  in a Lagrangian reference frame and calculates the trajectories of these particles using Newton’s second law of motion (equation of particle motion) on an individual scale. The particle momentum can formulate as18;Once the simulation of the flow field has been completed, the particle tracking technique applied. In this technique, STAR CCMThe drag force can compute as;from Eq.  and \docThe gravitational force formulated as;19:Erosion modeling predicts the rate of erosion from particle impact on solid boundaries. In calculating the erosion rate, STAR-CCM + accumulates the damage that each particle impacts cause. This calculation is done by selecting a correlation for the erosion ratio, that is, the mass eroded from the wall per unit mass of impinging particles. Erosion rate is computed by accumulating the damage on contact boundary faces that each solid particle impact does on the face20The Oka correlation for the erosion ratio isWhere: • The angle function • • ••••. Figure + code to generate the entire computational domain of the elbow. The mesh generation structure is shown in Fig. A three-dimension, multi-blocks, hexahedral and polyhedral meshes were created using the mesh generation section of STAR CCM17. The relaxation Gauss–Seidel scheme is applied to solve iteratively the generated set of linear algebraic equations to calculate the values of velocity, pressure, etc. A time step of ~ 1 × 10–4 s. was used for all calculations. Airflow field simulation was run for a 24-time cyclea to reach stable statistics results of the flow field and turbulence quantities. The solution procedure continued until reach the error criterion of for every component to satisfy the solution convergence criterion and Fig. The Eulerian-Lagrange simulation performed within the STAR CCM + firstly solves the system governing equations discretized by the finite volume technique along with using the SIMPLE algorithm method. In conjunction with this, the Menter k– SST RANS turbulence model employed to predict the turbulence effects on the airflow field The tested samples were 4 samples, one without coating and three with different coating thicknesses 30, 40, and 50 μm which are gotten from different periods of coating time 10, 20, 30 s, while the time coating more than 30 s (more than 50 μm) will give an unstable coating layers due to the lack of adhesion between the sample surface and coating layer as shown in Fig. Wear test based on weight loss, which was conducted on samples in the laboratory to compare between the wear resistance of coated and uncoated samples and simulate the result from the experimental test and its conformity. This test was done with (24) samples, through which a comparison made between samples without coating, (6) samples coated with a thickness of 30 μm, (6) samples coated with a thickness of 40 μm, and finally (6) samples with a 50 μm coating, the samples were weighed before the test by an exact digital balance (4 digits), The time to complete the test for each sample was (10 min). The Fig. The Fig. The thickness of the coating layer of the samples was calculated at different coating times by SEM Images and the measurement of the coating thickness was done by the image J soft program. It was found that the layers thicknesses of 30, 40, and 50 µm were achieved for coating times of 10, 20, and 30 s, respectively as shown in Fig. The SEM images showed also the coating time more than 30 s (thickness layer more than 50 μm) will produce an unstable coating layer that subjected to cracking, flacking, and failure due to lack of adhesion between the sample surface and coating layer as shown in Fig. Figure Figure Erosion that involves the removal of material from the surface of a component by the high-speed impact of a liquid or of a stream of hard particles carried in a fluid flow. The mechanism of erosion occurs when a stream of hard particles is directed at a surface. This may be intended, as in shot blasting processes, or it may arise incidentally, such as in pipelines and associated components carrying slurries or crude oil containing sand same as pipes and elbows transporting cement powder. This work aimed to find a unique technique to reducing this defect by using Nanoparticle coating with tungsten carbides (WC). This coating technique gave an improvement in wear resistance towards the cement particles flow by 71% for the sample with 50 μm coating thickness that tested inside elbow under same real operation conditions to get a vision about the behavior of coated and uncoated samples inside the elbow is facing the high-velocity cement particles flow. The coating layers can recoat after its removal many times. Also, the improvement in wear resistance for the 50 μm coating thickness sample that was tested by a specific standard test of wear (pin on disc) was 97% higher than samples without a coating layer. The result of the pin on disc test indicates the effect of the Nano coat layer which prevents the steel surface from degradation and increasing the operating life of pipes and elbows. The coating layer thickness more than 50 μm will subject to cracking, flacking, and failure due to the lack of adhesion with the steel surface. The numerical simulation that was done to compare with the test inside the elbow shows a good agreement between the experimental results and numerical results with a convergent reach to 90%.The failure due to mechanical wear in pipes and elbows may cause a lot of losses in money and times for industrial foundations and this failure occur due to