R2 = 0.9328 (p < 0.001). A significant increase in the residual error was found when each of four factors was excluded from the contributory variables. The study demonstrated the importance of four circulation factors in the prediction of the number of MBs and its capacity to eliminate potential postsurgical complication risks.The need for the estimation of the number of microbubbles (MBs) in cardiopulmonary bypass surgery has been recognized among surgeons to avoid postoperative neurological complications. MBs that exceed the diameter of human capillaries may cause endothelial disruption as well as microvascular obstructions that block posterior capillary blood flow. In this paper, we analyzed the relationship between the number of microbubbles generated and four circulation factors, i.e., intraoperative suction flow rate, venous reservoir level, continuous blood viscosity and perfusion flow rate in cardiopulmonary bypass, and proposed a neural-networked model to estimate the number of microbubbles with the factors. Model parameters were determined in a machine-learning manner using experimental data with bovine blood as the perfusate. The estimation accuracy of the model, assessed by tenfold cross-validation, demonstrated that the number of MBs can be estimated with a determinant coefficient  Dementia in the elderly and severe cerebral dysfunction  in children with congenital heart disease have been reported as the major types of complications caused by cardiopulmonary bypass5.The complications associated with cardiovascular surgery remain a major challenge as they can cause serious health complications10. Correspondingly, microbubbles (MBs) generated during the surgical field suction processes constitute a contributory factor of gas embolization. MBs are generated when blood and a large air volume flow together in the venous reservoir and are not completely defoamed12. A filter is commonly used in the venous blood reservoir and the oxygenator to protect the patient by removing inflowing air. However, it is difficult to prevent all the air that was transformed to MBs from being directed to the patient. Several studies have been reported wherein air sucked in the venous reservoir breaks down when it passes through the filter and separates into many MBs (10–20 μm)14. The MBs to be measured can be characterized in terms of quantity and size. Capillary destruction, obstructive ischemia, endothelial damage and activation of leukocytes and platelets are attributed to their numbers and sizes15. The diameters of capillaries range from 4–9 μm16, so that the number of microbubbles larger than this size may cause postoperative injury by being delivered. The MBs in the liquid move slowly and are delivered to the oxygenator by the negative pressure of the centrifugal pump used in cardiopulmonary bypass. The MBs delivered to the oxygenator are partially trapped, but the rest of them are sent to the patient. The number of trapped MBs depends on the type of the oxygenator. The current cardiopulmonary bypass system is equipped with a bubble sensor that can detect and warn for the presence of air bubbles with diameters ≥ 300 μm17, but MBs < 300 μm can potentially be sent to the patient. In this regard, current devices cannot adequately prevent the delivery of MBs.Advances in cardiopulmonary bypass systems have reduced complications associated with cardiovascular surgery. However, cardiopulmonary bypass surgery has been responsible for severe brain dysfunction primarily attributed to gas embolism22. These factors were identified in studies that aimed to a) reduce the number of delivered MBs and b) improve the defoaming process by implemented devices with improved performances, such as oxygenators24. The response characteristics of MBs with respect to the contributing factors were also described25. For example, the management of the venous reservoir levels has been proposed as an approach to minimize MB delivery. Kaza et al.19 and Rubens et al.20 have respectively reported a fine adjustment pump flow method to draw blood in the surgical field and another method to separate MBs with a separate reservoir. Chung et al.21 and Souders et al.22 reported that blood viscosity is related to the buoyancy of air bubbles, such that the slow MB floating in blood. They also reported that the force of the MB delivery is related to perfusion. Collectively, these studies revealed a close relationship between each factor and MB delivery. However, a comprehensive relationship between the four factors and MB delivery has not been determined yet.The four factors reported to be responsible for the delivery of MBs from the venous reservoirs during cardiopulmonary bypass surgery include the surgical field suction flow rate, venous reservoir level, blood perfusion flow rate, and blood viscosity26. Because the three factors (excluding blood viscosity) can be measured according to the output of the cardiopulmonary bypass, we are now able to examine the comprehensive relationship between the four factors and MBs. Blood viscosity repeatedly increases and decreases during cardiac surgery using cardiopulmonary resuscitation. Therefore, it is worth to monitor continually the viscosity and analyze its relationship with the number of MBs as well as other factors.One of the difficulties encountered was that the conventional cardiopulmonary bypass system could not conduct continuous blood viscosity measurements. In response, we developed previously a method that allowed real-time monitoring of continuous blood viscosityIn this paper, we model the relationship between the number of MBs generated from venous reservoirs and the factors associated with them. First, we performed a perfusion experiment with bovine blood as the perfusate. At the same time, the values of the four factors were measured in real time, and the relationships between each factor and the number of MBs were examined. The general expression for the relationship of the number of MBs was then derived with the use of a three-layered feedforward neural network. The parameters of the model were optimized with the measured values of the four factors. The estimation accuracy of the model was validated using tenfold cross-validation. Additionally, the contribution of the continuous blood viscosity was evaluated based on the comparison of the estimation accuracies of the models constructed with and without continuous blood viscosity as one of the tested variables.The distinctive feature of our method is the inclusion of the continuous blood viscosity that can be measured continually for the first time by the system developed by our group as a factor for the clarification of the relationship between the number of microbubbles generated and the four factors. Elucidating the relationship between the number of microbubbles generated and the factors associated with the number of MBs will enable us to identify factors that can be clinically varied and used to reduce the number of MBs. Among these factors, the suction flow rate of the operative field and the level of venous blood reservoir are factors that may be modified. Blood viscosity can be intervened in certain situations, although this has clinical limitations. This study elucidates the relationship between the number of MBs and related factors and proposes a neural network-based model that can be used during cardiopulmonary bypass surgery.r|≤ 0.7, p < 0.001). The distance to the reservoir outlet decreases as the venous reservoir level (VRL) decreases. The MBs thus immediately reach the reservoir outlet and flow out, then increasing their numbers. Therefore, a negative correlation was observed. In the case of the surgical field suction flow rate (SFR), the number of MBs flowing into the reservoir increases with the increase in the suction pump amount. In the case of blood viscosity (V), the number of MBs increases owing to the turbulent flow of blood in the vein reservoir and the decrease in the buoyancy of the incoming microbubbles as the blood viscosity increases. Blood perfusion flow rate (Q) increases the number of microbubbles by increasing the negative pressure that draws blood from the venous reservoir. Thus, these three factors are positively correlated with the number of microbubbles. Moreover, there were no significant correlations between all the pairs of factors. These results indicate that each factor may independently affect the occurrence and transmission of MBs.To understand the relationship between the numbers of MBs delivered from the venous reservoir during cardiopulmonary bypass surgery and the four factors , we performed perfusion experiments using bovine blood. In the experiment, different values of the four factors were configured to represent different perfusion conditions. At each condition, the numbers of MBs delivered from the venous reservoir were measured by a pulsed ultrasound Doppler system . The measurement time was 5 min at each condition. The numbers of measured MBs were represented by the median values in 1 s intervals. A correlation analysis was conducted between the four factors and the number of MBs, and results are summarized in Table Figure Considering the nonlinearity in the relationship between MB and the four factors, the general expression used to show the estimated the number of MBs was derived from a three-layered feedforward neural network see Fig. , and is entclass1pt{minimat in Eq. .1\documeR2 = 0.9328, p < 0.001). This result is a plot of the values obtained from the mean of the tenfold cross-validation. Figure R2 = 0.1674, p > 0.05). Figure R2 = 0.7131. Figure p = .Figure R2 = 0.3881, 0.5308, 0.7130, and 0.7742, respectively. The estimated accuracy decreased in the following order: SFR, VRL, V, and Q. This result indicates that it is effective to estimate the number of MBs using all four factors. Figure 27 was used for the evaluation. The root-mean-square-error (RMSE) was used as an index, and the mean value of the RMSE was obtained by the tenfold cross-validation. The RMSEs of the surgical field suction flow rate, the venous reservoir level, the blood viscosity, the blood perfusion flow rate, respectively worsened by approximately 14.1, 11.6, 7.3 and 5.7% compared with the baseline, when random permutations were conducted on each input. The result indicates that the surgical field suction flow rate is the most effective input, and the blood perfusion flow rate was the least effective in the estimation of the number of microbubbles.It should be noted that the estimated accuracies when one of the four factors was excluded  from Eq.  were R2 22 that were the surgical field suction flow rate, the venous reservoir level, the blood viscosity and the blood perfusion flow rate. Correlation analysis indicates that the four factors were not significantly correlated with each other, but were significantly correlated with the number of MBs  have been poorly documented32. Although the perfusate employed in this study was bovine blood, the reported experimental data may provide new evidence regarding the relationship between blood viscosity and the numbers of MBs generated by non-Newtonian fluids.The effects of blood viscosity were minor compared with the surgical field suction and venous reservoir level. By contrast, given that these effects are more prominent than those associated with blood perfusion flow, it can be an important factor. Based on Newtonian fluid experiments21. Hence, it was considered that these factors have led to an increased number of transmitted MBs. Therefore, measuring blood viscosity in real-time is necessary to estimate the MB delivery accurately. Souders et al.22 reported that an increase in blood perfusion reduced the blood transit time in the venous reservoir and increased the MB delivery even though the effect was small compared with other factors. This observation was consistent in all our experimental results  commonly obtained from cardiopulmonary bypass systems, the proposed method could be easily implemented. By using the proposed system, it is possible to capture the increase in microbubbles. Given that no single factor has a prominent effect on microbubbles, the complexity of the relationship among the four factors makes it possible to predict the expected number of microbubbles. MBs sent from the venous reservoir can be reduced by adjusting one of the four factors based on the conditions during operation. Therefore, for example, the proposed model can be applied to the development of a system that can present optimal suction conditions within the surgical field.In this study, increased estimation accuracy was achieved only when the four factors were incorporated as the contributory variables. This result indicates the importance of continuous blood viscosity that first became measurable with our previously published methodThe safety level of the number of microbubbles has not been determined. Thus, it is necessary to address this issue in the future. The parameters are determined based on the continuous operating conditions. It is impossible to detect an increase in the number of delivered MBs owing to blood transfusion and drug administration during surgery. Additional studies are required to construct a system based on this model to identify the optimal conditions for the execution of these procedures. Also, the number of MBs may vary depending on the venous reservoir used. Finally, the test fluid used in this study was mixed with bovine blood and Ringer's solution, so that a different result may be derived when human blood is used.The number of MBs generated during cardiopulmonary bypass surgery was increased or decreased owing to the complex combinational effects of the four factors that are the intraoperative suction flow rate, the venous reservoir level, the continuous blood viscosity and the perfusion flow rate. The proposed model can estimate the number of MBs by using a neural network-based regression approach. By clarifying the relationship between the number of MBs and related factors, we were able to estimate the expected number of generated MBs with increased accuracy.Figure The blood reservoir volume of the HSVR was 4000 ml and included a 40 μm internal screen filter. For this reason, microaggregates, such as blood clots and fat clumps that may be sucked from the surgical field, were removed. The bubbles were structured for treatment with a polyurethane defoamer. The minimum reservoir level of the HSVR was 150 ml. In the measurement of the venous reservoir level, the hydrostatic pressure in the reservoir was continually monitored.The membrane oxygenator used in the experiment had a built-in arterial filter that contained a screen mesh filter with a pore size of 32 μm adhered to the outside of the fiber layer. For this reason, microaggregates, such as bubbles, blood clots, and fat clumps were removed. The number of MBs was measured using a pulsed ultrasonic Doppler system  to detect the outflow of MBs from the HSVR. This system was set to detect MBs in the range of 10–250 μm. The probe was attached to the tube with an ultrasonic gel to eliminate air. Probes were reused in all experiments. The probe was mounted below the patient reservoir to exclude air which flowed out from the patient reservoir.26. This system is capable of measuring blood viscosity in real time by implementing a blood viscosity estimation model based on the characteristics of the fluid that perfuses and oxygenates the membrane. The model enables calculation of the blood viscosity from the pressure and flow rate information acquired from the cardiopulmonary bypass device through a USB port. The system then displays the estimated viscosity value and its trends as well as the pressure gradient and flow rate on a computer screen to facilitate the users for monitoring the viscosity-related information. The blood viscosity was adjusted to 3.0, 2.5, 2.0, and 1.5 mPa s based on blood temperature changes. MB measurements were initially conducted at a blood viscosity of 1.5 cp and were measured at all the combinations of reservoir level, suction flow rate, and perfusion flow. After the measurements of each combination were completed, the temperature was lowered to adjust the blood viscosity. MB measurements were performed when the MB transmission volume became stable after the experimental parameters were changed. The sizes of the MBs measured with BC-100 ranged from 10 to 400 μm. The MBs were counted regardless of their sizes. The numbers of MBs were expressed as the median values of each 1 s internal. Measurements were conducted in 5 min intervals. Furthermore, CO2 flushing is a technique used to remove residual air in the heart. In this experiment, the gas was also blown around the air suction at 3 l/min.Test fluid was prepared with 1000 ml of bovine blood and 1000 ml of Ringer's acetate solution so that the hematocrit was in the range of 24–26%. Hematocrit was not changed until the end of the experiment. The blood temperature was adjusted to 20 °C, 25 °C, 30 °C, and 35 °C, to change the blood viscosity. MB measurements began when the continuous blood viscosity became stable Experiments were performed subject to the different combinations of the experimental parameters. The rotational speed of the centrifugal pump that determined the perfusion flow rate was fixed at 2500 revolutions per minute, and reflux was performed at 4.0, 3.5, 2.5, and 2.0 ml/min based on the adjustment of the occluder. The reservoir HSVR levels were adjusted to 1000, 700, 500, 300, and 200 ml. The pump flow rates that emulated the air suction in the surgical field were 1500, 1300, 1100, 900, 700, 500, and 300 ml/min. The flow rate of the suction blood was 400 ml/min. The flow rate of 400 ml of suction blood was determined in consideration of the bleeding from the surgical field, the sampling line flow rate, the suction blood from the vent, and the hemoconcentrator flow rate. Each parameter was set within the range of the usual settings implemented in our system of the Hiroshima University Hospital. It should be noted that the flow rate of the system's surgical field suction can be set higher than that of the experimental conditions used in this study. However, the system is rarely operated in this setting, and even when it is set, it is used for a very short time. Thus, the settings for continuous operations are within the scope of this experimental method. Blood viscosity was measured continually during the experiment with the method we reported previouslyFirst, the correlation analysis between each factor was performed by JMP14  using the residual maximum likelihood estimation method. We then constructed a model with a multilayer neural network to enable the estimation of the numbers of MBs attributed to the four tested factors . The multilayer neural network model was implemented with JMP14. The neural network model consisted of an input, a hidden, and an output layer. Figure


In recent papers, Kenyon et al. (Ergod Theory Dyn Syst 32:1567–1584  A is called non-negative if all the entries are non-negative real numbers and aperiodic if there exists nth power We first recall a classical result for matrices dating back to work of Perron (1907) and Frobenius (1912) .A be a non-negative aperiodic (Perron–Frobenius Theorem) Let B with 0–1 entries such that We next recall a generalisation of the Perron–Frobenius Theorem to Banach spaces of functions. Let Let The following result of Ruelle is a cornerstone of the classical theory of thermodynamic formalism (cf.).There exists any the dependences (Ruelle–Perron–Frobenius Theorem) Let It is also known that, aside from the maximal eigenvalue pressure of the function The value ent}ϕ∈Fθ . In the Recently, several authors introduced a particular non-linear version of Theorem B be a non-negative irreducible  Let A has entries which are natural numbers, this appears as Lemma 1.2 in [In the special case that a 1.2 in . A versia 1.2 in ) under va 1.2 in  and 8]..A has enThe following is our main result, which can be viewed either as a non-linear version of Theorem There exists for any the dependence (Main Theorem) Let The result easily generalises to In the particular case that the function Let mentclass2pt{minimIf I am grateful to the referees and the editors for their patience and help with this short note.Although our main result Theorem  is of inFollowing  and 2, , 2, 3 gin; and We can define the pointwise dimension of rtitions .The pointwise dimension is particularly useful in estimating the Hausdorff Dimension of sets  a Fan–In this case one can consider the matrix The proof of the existence of the fixed point is the more interesting part of the problem. The uniqueness and analyticity are then relatively easy to establish.x and The existence can be shown by looking for a fixed point of a suitable map in the spaceWe can now introduce a family of non-linear operators defined as follows:For each We have that Let c is sufficiently large, Lemma By definition of We can use the above lemma to deduce the following.For c is sufficiently large that Let This now brings us to the existence of the fixed point for each of the operators For each By the Arzela–Ascoli Theorem the space We can now use the compactness of A). However, this contradicts that To see that Assume for a contradiction that we had a second distinct non-trivial positive fixed point, i.e., Interchanging the roles of For the definition of This simple argument doesn’t rule out the possibility of another non-positive fixed point.To show the analytic dependence of the solution we want to use the implicit function theorem applied to the function We recall the following result  which is(Ruelle) The operator In particular, we see from Lemma In particular, by  we see t(The Tangent Operator) Closely related to this circle of ideas is the use of a standard technique in understanding the iterates of a non-linear operator in a neighbourhood of a fixed point. More precisely, we consider the first order approximation to The classical transfer operator Given a solution (Variational Principle for Let and thus  follows We consequently have a particularly simple expression for the entropy We can write


The Earth’s magnetic field has operated for at least 3.4 billion years, yet how the ancient field was produced is still unknown. The core in the early Earth was surrounded by a molten silicate layer, a basal magma ocean that may have survived for more than one billion years. Here we use density functional theory-based molecular dynamics simulations to predict the electrical conductivity of silicate liquid at the conditions of the basal magma ocean: 100–140 GPa, and 4000–6000 K. We find that the electrical conductivity exceeds 10,000 S/m, more than 100 times that measured in silicate liquids at low pressure and temperature. The magnetic Reynolds number computed from our results exceeds the threshold for dynamo activity and the magnetic field strength is similar to that observed in the Archean paleomagnetic record. We therefore conclude that the Archean field was produced by the basal magma ocean. Cooling of the iron core in the early Earth may have been too slow to allow for the generation of a magnetic field. Based on quantum mechanical and geodynamical modelling approaches, the authors find that the electrical conductivity of silicate liquid at high pressure and temperature conditions could have been sufficient to generate a silicate dynamo and a magnetic field in the early Earth. The process by which the field is produced today: a dynamo hosted in the Earth’s metallic iron-rich liquid outer core is thought to be difficult to sustain in the early Earth because the core could not cool sufficiently rapidly4. Alternative energy sources, including radioactive heat production in the core or exsolution of Mg from the cooling core, may not be sufficient to sustain an early dynamo.The Archean magnetic field may have been essential for preserving the early atmosphere and hydrosphere5. A silicate dynamo is so far unknown. Magnetohydrodynamic simulations, laboratory experiments, and studies of planetary bodies that host magnetic fields show that dynamo activity requires a sufficiently large magnetic Reynolds number μ0 is the magnetic susceptibility, v is the flow velocity, L is the depth of the layer, and σ is the electrical conductivity7. Using illustrative values, 9, although less than typical metallic conductivity (106 S/m for the Earth’s outer core).Planetary dynamos in the solar system are hosted in a variety of materials including iron-rich liquid, such as in the Earth and other terrestrial planets, ionically conductive fluid ices, as in Uranus and Neptune, and metallic hydrogen as in Jupiter and Saturn2 and O systems11 (see Methods). Briefly, we compute the electronic contribution to the electrical conductivity with the Kubo–Greenwood formula12 and the ionic conductivity from the electric current autocorrelation function13. In order to investigate the effect of magnetism on our results, we consider non-spin-polarized and spin-polarized results with the latter characterized by high-spin moments on all the iron atoms. We combine our predictions of electrical conductivity with a model of the thermal evolution of the basal magma ocean to estimate the magnetic Reynolds number and the strength of the magnetic field generated by the silicate dynamo. We find that the magnetic Reynolds number exceeds 40 and that the field strength is in excellent agreement with paleointensity measurements.Here we perform first principles molecular dynamics simulations of a silicate liquid that closely approximates the composition of the bulk silicate Earth and includes the six most abundant oxide components; the size of our simulation is 1129 atoms (see Methods). The simulations and the computation of the electrical conductivity closely follow our previous work on the SiOσ exceeds the ionic contribution by more than an order of magnitude. This behavior contrasts with experimental results at low pressure and temperature where ions are the dominant charge carriers8.Our results show that the electrical conductivity of silicate liquid exceeds 10,000 S/m at the conditions of the basal magma ocean O liquid10 over the range of temperature and pressure that we have investigated. The conductivity of the silicate liquid is greater than that of SiO2 because of the role of iron. Iron 3d states contribute to the density of states at the Fermi level and increase the conductivity compared with the iron-free system. The role of iron also explains why the temperature dependence of the electrical conductivity in the silicate liquid is less than that of SiO2 liquid : the highly localized iron 3d states are less sensitive to their chemical environment than the O 2p states that also contribute to conduction and their contribution to the density of states at the Fermi level is much less sensitive to temperature. The conductivity of the silicate liquid is less than that of (Mg0.75Fe0.25)O because the iron concentration is less in the silicate liquid (Fe/(Fe + Mg) = 0.11). The electronic density of states also explains why the conductivity of the spin-polarized system is slightly less than that of the non-spin-polarized system: magnetism splits up- and down-spin iron 3d states, reducing the density of states at the Fermi level. While there are no experimental measurements of the electrical conductivity of silicate liquids at basal magma ocean conditions, we have shown that our approach yields results that are in good agreement with available experimental data at high pressure including the optical reflectivity of silica liquid and the electrical conductivity of crystalline O11.We can understand the electrical conductivity of silicate liquid by comparing with simpler systems and by examining the electronic density of states Fig. . The ele16, except that we have replaced the previously assumed value of the entropy of melting with the value determined from first principles molecular dynamics simulations (see Methods). This model yields the thickness of the magma ocean L and heat flux out of the magma ocean q, which we combine with our values of the electrical conductivity to estimate the magnetic Reynolds number ρ is the mean density of the magma ocean and HT is the thermal scale height. We have assumed the mixing length scaling for the flow velocity17. We estimate the magnetic field strength at the top of the basal magma ocean according to the scaling derived from numerical simulations17η = 0.1 Pa s is the viscosity of the basal magma ocean from previous first principles simulations18. Finally, we estimate the field strength at the surface rM is the radius at the top of the basal magma ocean and rE is the radius of the Earth6.To quantify magnetic field production, we combine our results for the electrical conductivity with a model of the thermal evolution of the basal magma ocean. The thermal evolution model is identical to that examined previously19 suggested as the criterion for magnetic field generation the following dynamo number Rmcore = 3500 is the magnetic Reynolds number of the core computed assuming that the horizontal flow velocity is similar to that of the core today. According to the dynamo number criterion, the silicate dynamo may operate throughout the Archean . It is likely that the basal magma ocean became more iron-rich as it crystallized and that this iron enrichment increased the electrical conductivity. Our results for Rm and D may therefore be underestimates. We have not attempted a more quantitative treatment of the effects of iron enrichment because liquid-crystal iron partitioning and the magnitude of the effect of iron enrichment on the electrical conductivity of silicate liquid at basal magma ocean conditions are uncertain.Our results show that a basal magma ocean is likely to have produced a magnetic field Fig. . The mag20 and the precipitation of Mg21. However, recent experimental studies show that the solubility of K and U in the core is too small to power a dynamo23. Moreover, a recent reanalysis of experimental data shows that the temperature dependence of Mg precipitation, and thus the power produced, is a factor of six less than originally thought24 and may be less than that required to sustain a magnetic field25. The field produced by the silicate dynamo may have differed in geometry from the present day field and it may be possible to distinguish these unique characteristics in the geologic record. For example, there is some evidence from modeling magnetic field generation in other planetary bodies, including Uranus and Neptune26, that shallow layers, like the basal magma ocean, produce a less dipolar field than the strongly dipolar field of the present Earth. Many super-Earth exoplanets may have extensive magma oceans because of the greater amount of accretional energy stored in larger mass planets27, strong stellar irradiation, or the insulation of overlying gaseous envelopes28. Silicate dynamos may therefore be common in the universe.The Earth’s magnetic field may have transitioned from being produced by the basal magma ocean to being produced by the core near the end of the Archean: as the magma ocean cools, heat flow out of the core increases to the point that core convection, and a core dynamo, is possible O, differences in σ between simulations containing 128, 256, and 512 atoms were less than 10%.Our molecular dynamics simulations are based on density functional theory in the PBEsol approximation12 as implemented in VASP from a series of at least 10 uncorrelated snapshots at each volume–temperature condition  and a larger number of electronic bands . We compute the electronic density of states by averaging over 10 snapshots well separated in time from the molecular dynamics trajectories. We use the energy derivative of the Fermi–Dirac function with temperature equal to the ionic temperature to smooth the density of states.13zi and i, respectively, and the angle brackets indicate an average over time origins t0. The Bader charges of the ions computed in our simulation are listed in Supplementary Table We compute the ionic part of the electrical conductivity in the DC limit from the electric current autocorrelation function16:a, and its temperature TL where k is the thermal conductivity, TM is the temperature of the overlying solid mantle, δ is the thickness of thermal boundary layer at the base of the overlying mantle, M and c are the mass and isobaric specific heat, respectively of the basal magma ocean (subscript m) and core (subscript c), H is the radioactive heat production, ρ is the density of the basal magma ocean, and ΔS is the entropy change on freezing. The inner radius of the basal magma ocean b is assumed to be the core–mantle boundary. The term on the left-hand side is the total heat flux out of the basal magma ocean. On the right-hand side we have contributions from, respectively, cooling of the basal magma ocean and the core, radioactive heat production, and latent heat of freezing. The second equation expresses the idealized phase diagram defined by a linear dependence of the liquidus temperature on the mass fraction of the dense component ξ, where TA and TB are the melting temperatures of the end-member light (A) and dense (B) components. The last equation is a statement of mass balance where Δξ =  ξL − ξS is the enrichment of the liquid in the dense component. The model does not include inner core growth, nor radioactive heating, nor Mg-exsolution in the core.We solve the coupled system of equations 16, except for the entropy of melting, which we take from our previous simulations ΔS = 652 J/kg/K37  forces: CIAMACTo estimate the flow velocity, we use the scaling recommended by numerical simulations19 suggested as the criterion for magnetic field generation the following dynamo number D: the geometric mean of the magnetic Reynolds numbers of the basal magma ocean and the coreRmcore = 3500 is the magnetic Reynolds number of the core computed assuming that the horizontal flow velocity is similar to that of the core today, and D > 100 in order to produce a magnetic field. Following Ziegler and Stegman, we have estimated Rmcore by assuming that the horizontal flow velocity is similar to that of the core today (v = 5 × 10−4 m/s)38, the conductivity, σ = 1.5 × 106 S/m39, and L = 3500 km.We also estimate the possible effects of coupling of the magnetic field produced by the basal magma ocean with the underlying, non-convecting core. Although the basal magma ocean suppresses core convection, the core is likely to have strong horizontal flows that generate toroidal fields from the poloidal field produced by the basal magma ocean, reinforcing the silicate dynamo. Ziegler and StegmanSupplementary Information


The results show that the drop elongates in the direction of the field with an increase of its intensity. As soon as the field strength reaches a certain value, the interface and the free surface of the drop become unstable, which causes the peaks of different height to form. It has been found that the ratio of the corresponding critical values of magnetic field intensity is determined by the ratio of surface tension at the interface to that on the free boundary of the drop. Surface and interfacial tension of liquids used in the experiment were measured with the help of tensiometer by the ring detachment method to verify the experimental data. The presented results on the ferrofluid interface tension measurements can be of interest for the specialists in the field of ferrohydrodynamics.  Direct measurements of surface tension of magnetic fluid being in contact with another nonmagnetic immiscible medium are still under the strong interest of scientists In this paper, the object of consideration (floating MF drop) changes its form from an almost flat free surface on the border with the air on one side, and the semi-ellipse at the interface on the other side to the peak instability at both sides. The proposed method is based on the ratio of critical field intensities, so the surface tension could be determined regardless the demagnetizing factor. Another distinguishing feature of this research is the use of direct measurements of the MF surface and interfacial tension with the help of certified tensiometer (Sigma 701), in order to verify the proposed method.0 and saturation magnetization MS, due to the particle sizes d of the solid magnetic phase, the phase's volume concentration φ and the average magnetic moment <m> of the particles In the experiment, three samples of kerosene-base ferromagnetic fluids, hereinafter called FF 1, FF 2 and FF 3, of equal density (ρ = 1.38±0.02 g/cm³) were used. These samples are also characterized by: different initial magnetic susceptibility χ2) was placed on a horizontal thin platform between two Helmholtz coils of diameter 180 mm (1). The cuvette axis coincides with the axis of the coils. The cuvette was filled with perfluorooctane C8F18 (3) which served as a liquid substrate (ρ0 = 1.76 g/cm³) for a ferrofluid drop. The VESTA BM2202 electronic weighing scales were used to determine (with an accuracy of 0.01 g) the substrate mass and, accordingly, its thickness. To this end, a syringe with perfluorooctane was weighed before and after injection of its content into the cuvette. The depth of perfluorooctane substrate was fixed at 40 mm to prevent the FF drop interacting with the boarders. In the absence of a magnetic field the drop used to tend to the side wall of the vessel due to the gradient in the surface tension.A glass cuvette with a square cross section 59 mm on a side and 45 mm in depth  was used4) was placed on the substrate surface with a mechanical Biohit Proline pipette, which made it possible to determine the drop volume in the range of 5 to 50 μl; the accuracy was 0.1 μl. The volume of large-size drops generated by the syringe was determined by weighing them. The initial diameter of the drop suitable for the method, which, according to calculations, was The ferrofluid drop . All the experiments were carried out at an ambient temperature of (26 ± 1)°C.The strength of the magnetic field a).b), which is accompanied by the appearance of a peak at the liquid-liquid interface (c). Note that the curvature of the surface changes abruptly and this is indicative of a jump-like pressure redistribution. It should be mentioned, that the peak formation on the FF 2 drops’ surfaces appeared to be a time consuming process due to its low magnetic susceptibility. The critical values of magnetic field for FF 2 were defined for the moment of reforming of peak lateral surface from convex to concave.A gradual increase in the strength of the magnetic field causes the lower part of the drop to elongate until the field intensity reaches the critical value Hi* , b, whicnterface , c. Noted). At the next critical value of the field strength e–f). As the field strength decreases, the drop shape changes in the reverse sequence, although the disappearance of the peaks is observed at lower values of the magnetic field intensity.When the magnetic field strength increases further, the height of the peak also increases and the immersed part of the drop takes the form of a cone, which elongates along the direction of the field , d. At trrofluid , e–f. AsiH* and fH* weakly depend on the initial drop diameter d0 but strongly respond to the variation of the ferrofluid initial magnetic susceptibility (a). Moreover, iH* and fH* remain unchanged with decrease in the initial drop diameter to 0d ≤ 3 mm, when the surface forces begin to dominate over the volumetric forces (capillary length λc=(ρg/σ)1/2 for the examined ferrofluids bordering the air medium is about 1.5 mm). With an account of the latter restriction, it may be empirically obtained that0 = (2.0b) according to the The critical values of the field strength tibility , a. More.0÷16.5) , b accorfH*/iH* is independent of d0, we may suppose that fH*/iH* is a power function of the ratio of the surface tension at the interface σi to that on the free surface σf.Bearing in mind that the only mechanism inhibiting deformation of the drop in the magnetic field is the surface tension and for a single drop the ratio i and interface with perfluorooctane σf of the examined fluids were measured in the absence of any external magnetic field. The Sigma 701 tensiometry system , that gives at least the σ resolution of about 0.001 mN/m, was used. The ring detachment method To verify this supposition, it is necessary to determine the surface tension at the interface between the drop and the surrounding medium. However, according to As mentioned above, along the experiment the ferrofluid density remained unchanged due to addition of extra kerosene (which was different from that kerosene used for their preparation). Since kerosene is composed of carbon chains that typically contain between 8 and 15 carbon atoms per molecule The analysis of the tabulated data allows us to conclude that expression (2) reduces to the following empirical equationb) and calculations from the I, II, III  correspond to the values As a result of applying the fH*/iH* agrees well with the calculation data from formula (3), which opens the possibility of determining the surface tension at the interface between the magnetic fluid avoiding the use of tensiometry system.As follows from the comparison, the experimental value of ω(k) for ferrofluid cH values corresponding to the instability of FF surface are determined by the gravitational-capillary waves on it cH and σ is reasonable.The theoretical interpretation of these results could be given according to the following dimensional analysis The conducted experiment revealed a specific character of changes in the shape of a magnetic fluid kerosene-based drop, due to the presence of a liquid substrate of perfluorooctane. The fact that deformation can affect the entire surface of the drop leads to the development of instability at the interface between the drop and the surrounding medium. The following instability causes redistribution of the ferrofluid inside the drop between its upper and lower parts (relative to the interface boundary). It has been shown that the ratio of the critical values of magnetic field intensity is determined solely by the interfacial tension ratio to the free surface tension. This finding may be useful to elaborate non-contact techniques for the measurement of interfacial tension of kerosene-based magnetic fluids.The surface tension of a magnetic fluid under the action of a magnetic field is the relevant problem that actually could be much wider – the surface and interfacial tension of composite materials in the external fields. This method is surely a trial to resolve this problem. It could help one to get the result for rather low magnetic fields while the surface tension weakly differs from the stationary one. According to One more significant point is that the form of the object (FF drop) changes during the experiment in a magnetic field that provokes the irregular changes in the demagnetizing factor of the upper and lower borders of the drop. Still, due to the usage of the critical field values’ ratio the demagnetizing factors do not affect the results of the measurements.


The model was run for a number of realistic scenarios in high horizontal resolution (∼30 m) with sea cages influencing the flow field. Comparisons between extensive current measurements and the simulation results showed that the model system reproduced the current local current field well. By running simulation scenarios with and without the effects of the sea cages on the flow field, it was possible to assess the importance of such effects for numerical dispersal models and aquaculture environment interactions simulations and hence for assessment of environmental impacts.Most Atlantic salmon mariculture operations use open sea cages for the grow out phase. The ultimate fate and effects of the effluents and the possibilities of disease transfer between fish farms are major concerns for farmers, governance and the general public alike. Numerical model systems applied to studying and managing effluents and disease transfer in mariculture must realistically resolve the hydrodynamics in the vicinity of the fish farms. In the present study, the effects of the aquaculture structures on the current patterns were introduced in the ocean model system SINMOD. The drag parameters for the ocean model were determined by comparing the simulation results from the ANSYS Fluent  Numerical simulation models are standard and indispensable tools in studies and assessments of environmental effects and interactions of open water aquaculture systems , 2. SomeSalmo salar) farms, but without considering the farm structures 17].Water currents were measured at six points around the farm at Rataren  for two Eulerian passive tracers were used to investigate the potential cage structure effects on dispersal of dissolved matter from the farms. A constant release of 1 unit PT per net pen per hour was used. The tracer was released homogeneously from the surface to the bottom of each net pen, assumed to be 25 m deep. Simulations were run with and without taking into account the effects of the cage structures on the water currents, for both of the fish farms Ørnøya and Rataren  and for Though passive, i.e. not equipped with any half-life or degradation rates, the tracers can be interpreted as dissolved nutrients like ammonium or phosphorous, pharmaceuticals, and as such will provide an upper estimate for concentrations and potential for water contact.® and SINMOD simulations for u0 = 0.15 ms-1 are shown in δ = 4 × 10−5 for the SINMOD results. The CFD simulations indicate a reduction of the current speed at 10 m depth down stream from the farm with a fraction of 0.44, while the corresponding fraction in the SINMOD simulations was 0.46.The results from the idealized Fluentδ. Corresponding curves for each of the entry current speeds u0 and each depth  are also shown. The minimization by  that is relevant here.In a 32 by 32 m model grid cell, a fish net pen structure accounts for a small fraction of the volume only. The volume of a net pen of 50 m diameter, 25 m deep, is approx. 36,000 mAccording to the current measurements, the main flow direction was SW-NE at all stations Figs –10, exceAt ADCP1 (results not shown) the observations and simulations had similar current speeds, but the simulation results were a lot less spread out in direction than the observations. The site is very shallow and fairly close to land, and is a position where it is difficult to reproduce the situation in a model. There may also have been more noise in the observations at this sampling stations than at the others due to the shallow bottom depth.The observations and simulation results were in agreement also at the MIDI buoy .An analysis of the current observations confirmed that the tide in the region is semi-diurnal with M2 the dominating component .The effects of the fish cages was not very strong at either of the sampling stations Figs –10, and During periods of more or less unidirectional currents through the farm Figs  and 11, At both farms  the passThe dispersal at the two farm sites was quite different. At Rataren, the dispersal of the PT seemed to be mainly along the SW-NE-axis Figs  both in The dispersal was greater in May than in August, at both fish farms, in particular when the effects of the cages were included Figs  and 14.The region covered by high (≥0.1 units) PT concentrations was greater at Ørnøya than at Rataren for both simulation periods. This reflects the facts that the amounts of PT released from Ørnøya were greater than at Rataren (12 vs 10 cages) and that the physical dilution was greater at Rataren (higher current speeds).While the average dispersal patterns were different in August-Sep, 2015 and in May-June, 2016, the differences between the scenarios with and without the cage effect was greatest in May-June at both farms , higher concentrations in some points necessarily imply lower concentrations in others. This is seen at the 1000 and 2000 m samples at Rataren and at the 2000 m samples at Ørnøya , where tt Ørnøya . There wThere were indications that the passive tracer concentrations released from either of the farms and sampled at the other  changed significantly  by introducing the cage effects on the current field. Concentrations of the tracers released from Rataren sampled at Ørnøya were generally higher than the concentrations of the tracers released at Ørnøya sampled at Rataren.This paper considers important aspects of and processes related to simulation of currents and dispersal of dissolved matter from open water aquaculture installations. Current measurements support the simulation results from the ocean model (SINMOD). In an idealized setting like a channel, the effects of the farm structures in SINMOD qualitatively match those of the CFD simulations, but in reality or in a realistic model setting are more difficult to confirm or reproduce . HoweverOn a qualitative level, the agreement between the simulation results for Rataren and the current measurements was good. In particular, the model reproduced the current directions well. In some cases the current speeds were too high . In other cases the spread of direction was a lot narrower in the model than the observations. At one station (ADCP1), the observations were spread out in almost all directions, while the simulations indicated currents in SW/NE direction. However, this observation point was very close to land and the bottom depth was less than 10 m, probably causing some of the differences here.The direct correlation between simulated and observed current time series was not very good. This is quite common in ocean modelling and is the reason why studies such as this use the longer term statistical properties of the current dynamics rather than time series correlation directly to assess the skill. Despite this, ocean models can fairly accurately estimate properties such as current speed and directions. In the present study, the current roses from the observations and the model were similar, indicating that the current patterns and dispersal over time scales of days and weeks were reasonable. Except for special and critical short-term operations, it is mostly from such a time scale and upwards that simulation results of this kind are being applied in management and assessment systems.δ between 0.03 and 0.05 we would potentially obtain a lower RMSD in δ = 0.04 compared with other regions (δstruct.By using a higher resolution in the selection of candidate parameters  regions . Therefoδstruct by ®).This conclusion is based on the simple definition of The use of CFD modelling for parametrization of the ocean model is a new and efficient approach. It can be seen as a way to link processes on different temporal and spatial scales. The fact remains, however, that the ocean model resolution prohibits resolving all local features in the hydrodynamics around aquaculture sea cages. Previous efforts have included parametrization based on laboratory experiments .The depth profiles of simulated currents were qualitatively similar to the results presented in , and shoThe simulated depth profiles  also indThe percentage reduction in current speed through the cage in the channel setup  was compThere were no current observations available from the Ørnøya farm, so no case comparison between observed and simulated values is possible here. The Ørnøya farm was included in the study because it demonstrates the potential for ocean modelling as an exploratory and not only prognostic tool. The Rataren and Ørnøya farms were in contrast to one another in terms of tracer concentrations, and there is also the dimension of water contact between the farms which is relevant for management systems for disease transfer.For dispersal of organic matter in a “near field” context (up to a couple of farm diameters), the results on the current modification and tracer concentrations are both interesting. Previous work has indicated that current modification due to farm structures impacts on simulation of the potential for deposition of organic matter . Becausepractical significance in this context.Our results are also interesting in a “far field” context. Firstly, transport models indicate a potential for “far field” dispersal of organic matter from fish farms  that hasThe ocean model SINMOD resolved the current fields realistically at and around the farm at Rataren. While the effect of the fish cages was not obvious from the current fields alone, by measurements or simulations, this effect manifested itself clearly in terms of passive tracer dispersal simulations. Thus, the inclusion of cage effects in aquaculture environment interactions simulations should be considered as these effects impact directly on dispersal simulations and hence the concentrations of effluents in both the near and far field zones. There were both temporal and spatial differences in how the cage structures affected passive tracer dispersal.


Legendre polynomials form the basis for multipole expansion of spatially varying functions. The technique allows for decomposition of the function into two separate parts with one depending on the radial coordinates only and the other depending on the angular variables. In this work, the angular function  The multipole expansion involves expressing a function as a linear combination of Legendre polynomials, or the related spherical harmonics, with the orders of expansion in this case being the orders of the poles in the multipole expansion 3. Many special functions, such as ordinary and spherical Bessel functions5, arise naturally as the radial part of a function and spherical harmonics as the angular part whenever a function is separated by the multipole expansion series.The multipole expansion is a powerful mathematical tool useful in decomposing a function whose arguments are three-dimensional spatial coordinates into radial and angular parts. This simplifies the solution of physical problems by reducing the triple integrals into a one-dimensional integral of the radial part and the two angular integrals. The angular integrals are solved using angular momentum algebra  6 is derived. The generalization is used for the multipole expansion of the plane wave In this work, l, with l a non-negative integer, are smooth functions defined in the region l while for odd values, it runs from one (1). The first two Legendre polynomials are 3The Legendre polynomials mentclass2pt{miniml and k. Similar to Eq. (k is even and from one (1) if k is odd. The first ten (10) Legendre polynomials3 are expressed in terms of In this work, an expansion is done in a reverse process by expressing mentclass2pt{minimmentclass2pt{minimx, and then similarly replacing the lower powers of x with the coresponding functions of Legendre polynomials. This implies that the reverse process has to begin with x in the expression for In the reverse process, mentclass2pt{minimk being a non-negative integer and where Our goal in this study is to obtain a multipole expansion of k and l respectively.Table sing Eq.  up to Eqsing Eq.  for the k for each sequence as:l. The results determined in this study for each sequence are presented in Table k and l.The next task involves forming the sequence of coefficients corresponding to each order of the Legendre polynomial and deductively determining the pattern with the considered cases. Noting that k and l. The derived expansion coefficients’ algorithm 6.Based on observations of the patterns of As a standard application, we consider the multipole series expansion of the plane wave, d in Eq. . LikewisThe conventional multipole expansion series of the Coulomb repulsion term, on the other hand, can be derived by rearranging the terms of the Binomial expansion as9,With the substitution of the series expansion of d in Eq. , and furd in Eq. , the sumd in Eq.  simplifid in Eq. . In the Our goal in this paper is to expand  in Eqs.  and 25)\document


Microfluidics exploiting the phenomenon of inertial focusing have attracted much attention in the last decade as they provide the means to facilitate the detection and analysis of rare particles of interest in complex fluids such as blood and natural water. Although many interesting applications have been demonstrated, the systems remain difficult to engineer. A recently presented line of the technology, inertial focusing in High Aspect Ratio Curved microfluidics, has the potential to change this and make the benefits of inertial focusing more accessible to the community. In this paper, with experimental evidence and fluid simulations, we provide the two necessary equations to design the systems and successfully focus the targets in a single, stable, and high-quality position. The experiments also revealed an interesting scaling law of the lift force, which we believe provides a valuable insight into the phenomenon of inertial focusing. With such a promising performance, the technology has grown rapidly.Inertial focusing is a phenomenon that enables focusing of initially randomly distributed particles in a fluid into well-defined positions within microfluidic channels, thereby facilitating the detection, isolation and analysis of rare targets of interest in complex fluid samples like blood, for instance. The technology has attracted much attention over the last decade thanks to its attributes; it allows for high through-put focusing, concentration and separation of particles with high resolution, it does not require labelling of the targets, it works for neutrally buoyant particles, and the operation of the systems is relatively simple (the sample simply has to pass through the microchannel at a controlled flow rate)9 since it was first observed by Segré and Silberberg in 196110. Migration and focusing of particles occur in microfluidic systems where inertia is not negligible, and it is attributed to a net lift force  from blood samples12 and focusing, separation and concentration of pathogenic bacteria from water samples14. Modern reviews covering the physics, the performances from different configurations and cross sections, and successful applications can be found in the literature18.The phenomenon has been physically and analytically described13. Although robust silicon-glass systems have been shown to withstand up to 200 bar and allow for focusing particles down to 0.5 µm14, smaller particles of interest such as viruses or exosomes remain out of reach. Another major limitation is the fact that the position of the focused particles (focus position) generally depends on multiple variables. In fact, the focus position generally shifts in tortuous ways as a function of all variables defining the system and the flow: the width  microchannels20, a line of inertial focusing that overcomes the major limitation of the shifting focus positions. The systems consist of curved rectangular microchannels with an aspect ratio A notable limitation for the technology is the rapid increase in the pressure needed to run the systems when targeting submicron particles21. The other extreme, consisting purely of Figure entclass1pt{minimaentclass1pt{minimaentclass1pt{minima20. Second, 17, there is no consensus for an accurate expression for To succeed in engineering a HARC system and achieve a performance like that shown in Fig. In this paper, we studied the upper limit experimentally. We gathered experimental data of Last, since the experimental measurements contain information about the strength of 23; Fig. 21). Although there is no agreement on an expression for In HARC systems, there is a net lift force  that takes the shape of two vortexes and drags particles in the direction of its streamlines20, and the retention/focus of particles at the inner wall; the focus of this paper.The study of inertial focusing in HARC microchannels can be divided in two sections: The collection of particles around the EP by the secondary flow, which was explained in detail in our previous work20 , which sets the lower limit of the systems. An approximation to the necessary channel length for this to happen, expressed as the number of loops , is then defined at the symmetry line and With this, the study of the balance between the two forces in HARC systems is reduced to a single point (ROI). Expressions for both ribe Eq.  and provThe devices were fabricated on hybrid silicon-glass systems so they could stand up high pressures without undergoing deformation. First, the microchannels were patterned on a silicon wafer with photoresist 1813 . The microchannels were dry etched with the photoresist as mask using a short cycle Bosch process to minimize the roughness of the sidewall . The wafer was then cleaned and 0.5 µm of Al were sputtered to cover the channels and act as an etch stop for the next step. Lithography was done on the back side  and via holes were dry etched for the fluidic connections . With all the micromachining finished, the silicon wafer, together with a borosilicate wafer, was cleaned and activated in piranha solution for 15 min. Both wafers were put together and anodically bonded (380 ºC and 1 kV for 4 h). Last, glass capillaries  that served as fluid connections were glued with epoxy (EPO-TEK 302-3M).Fluorescent polystyrene particles  were suspended in deionized water (with 0.1% of Triton X to reduce agglomeration) in a concentration of ∼ 0.001 vol%.An HPLC pump  was used to pump the samples through the devices at a controlled flow rate with a read out of the pressure.During the operation, the devices were observed with an inverted fluorescence microscope . Images were taken with a magnification of 20X and a 2 s exposure time. The intensity profiles were analysed by ImageJ.The conditions that lead to We mapped The results with this first set of devices reflected the influence of Simulations of the fluid flow were performed in microchannels with different cross sections using COMSOL Multiphysics v.5.5  in order to understand the secondary flow  and the roughness of the sidewalls by dry etching are the same in both cases, but the impact is much bigger for a small system. Equation  can therFinally, as it will be explained in the next sections, Eq.  can be ep of Eq. , obtainiWith Eq.  definingentclass1pt{minimaNote that Eq.  was obtaFrom a practical point of view, Eq.  expresseEquation  can be rentclass1pt{minimaBy analogy with Eq.  . Our experimental results here presented also align with the trend mentioned above. The obtained expression for The scaling of the lift force, here indirectly measured experimentally, points in a very interesting direction. The scaling was originally thought to be t}FL Eq.  can be r23 in 2013, where the channel length needed for particles to reach the equilibrium perimeter in straight channels was observed to be invariant with the flow rate up to moderate Furthermore, Finally, introducing Eqs.  and 10)10) into Re-organizing Eq.  to have With Eq. , we obtaregion of interest, ROI). Analytical expressions for the calculation of the forces (Eqs. To summarize, in this paper we identified a critical position of the cross section of HARC microchannels where the balance of the lift force and the drag by the secondary flow defines if the system focuses the particles or not (ces Eqs.  and 11 aces Eqs.  at said ces Eqs.  and, togces Eqs. , which wFrom a practical point of view, Eqs. –14 descr2.From a theoretical point of view, the experimental observations were related to the lift force and an expression revealing its strength and scaling was derived Eq. . While tThe Lift Barrier constant was obtained with water; the only fluid used in these experiments. The dimensions Last, HARC systems provide the means to focus a range of particle sizes together in a stable line; ideal for laser interrogation and concentration. However, precisely because of such a feature, the systems lose one of the key capabilities of inertial focusing, the possibility to separate particles. We are, nevertheless, working on solving this and we will present the results in the near future.With this work, the description of HARC systems for focusing particles is completed. The systems must be engineered to operate between two limits. The lower one was previously defined and ensures that particles have time to reach the focus position. The upper one, which is developed here based on experimental evidence, sets the limit where the lift force is no longer strong enough to stop particles from following the secondary flow and the system fails to focus them.Expressions for the design of HARC systems that allow for a high quality, single and stable position particle focusing are provided. Of special interest is the measured magnitude and scaling of the lift force, which may provide a valuable insight for the community.We believe that HARC systems, with an intuitive focusing mechanism and two simple equations to achieve a stable focus, make the technology of inertial focusing and its excellent performance easily accessible, which may facilitate its implementation outside research laboratories.Supplementary Information


Surface texturing is an indispensable way of increasing absorption in solar cells. In order to properly characterize the effect of texturing, the angular dependence of the incidence light should be addressed. This is particularly important when the actual application where the incidence angle of the sunlight varies during the day is considered. This study presents the angular dependence of solar cell parameters in the case of periodically textured crystalline silicon (c‐Si) solar cells with microholes. A standard solar cell with pyramid texturing is also studied for comparison. It is shown that the incidence angle for the highest efficiency depends on the surface structure. While a standard pyramid‐textured surface performs best at the zero angle of incidence, it is needed to tilt the sample with microholes textures 15° with respect to the surface normal. This is also confirmed by the simulation study performed for the structures presented in this study. The efficiency of a solar cell strongly depends on the properties of the interaction between the incoming light beam and the surface of the device. This study shows the angular dependence of performance in a solar cell with various surface texture patterns. It is concluded that for each surface pattern there exists a different tilt, resulting in maximum cell performance. For STC sunlight beam hits directly on top of the surface without any tilt angle relative to the surface. However, these conditions do not describe realistic situation; they only represent a small portion of actual conditions in which solar cells and panels operate in the application fields. In order to assess the performance of the solar cells under actual conditions, characterization techniques should be adopted depending on the medium of the device, the light intensity and the light incidence angle, physical location of the cell according to the duration of a day and climate. Standard photovoltaic modules are ground fixed, but the sun’s position is changing in time depending on earth’s movement, so an angle resolved performance analysis is the only way to obtain actual field performance of the system. This is particularly important when the surface of the cell is textured with a 3D structure such as pyramids, micro and nanorods, pillars, holes etc. Recently, a variety of such surface structures have been proposed and tested for a better light management on the surface of the solar cell. We have recently demonstrated fabrication of solar cell on Si wafer textured with periodic holes with micrometer (µm) size. In this case of a periodic hole structure shown in Figure However, for a better understanding the device performance under actual condition, angular dependence of the measurement needs to be addressed.The characterization of c‐Si is carried out under standard test conditions (STC) specifying an operation temperature of 25 °C, irradiance of 1000 W m surface.1, 2Jsc) values. This phenomenon is schematically illustrated in FigureWhen light enters the holes on the surface, multiple reflection from the walls of the holes leads to higher absorption and thus an increase in the short circuit current density Jsc valuesIn this paper, we report on the angular dependence of photovoltaic performance for devices having microholes as a texturing structure. We present both simulation results performed by SILVACO‐ATLAS program and experimental results. We show that for an accurate analysis of textured surface, the angular dependence should be taken into the account.1.1FigureIn this study, p‐type silicon wafers with <100> orientation having thickness of 525 µm and resistivity of 1–5 Ω cm were used as substrates. Microholes with varying dimensions and solar cell devices on these holes were fabricated by following the experimental procedures shown in Solar cells were then measured by an angle resolved measuring system where the incident angle was varied by tilting the sample in front of the collimated light of the solar simulator. In order to understand the measured angular dependence, we have also performed a simulation study using SILVACO‐ATLAS platform.2 Silvaco is a Device (ATLAS) and a Process (ATHENA) TCAD simulation tool, consisting of various modules to model semiconductor devices and their associated physical phenomena. Luminous (part of ATLAS) gives the absorption profiles for the device which are then converted into photogeneration profiles. The generation profiles are then used in the carrier continuity equations to calculate the IV‐characteristics of the solar cell. ATLAS provides four physical models for light propagation: Ray tracing (RT), transfer matrix method (TMM), beam propagation method (BPM), and finite difference time domain (FDTD) method. In our case, since the dimensions of the device are large compared to the wavelength of light, ray tracing model is the appropriate choice. RT ignores the interference and coherence effects in the propagation model. RT is computationally inexpensive, which makes it ideal for simulating thick solar cells.To simulate the solar cells with hole texturing, the commercial TCAD software Silvaco was used.17 SilModel Description: To properly model carrier transport in solar cells (carrier separation), it is necessary to incorporate Shockley Read Hall (SRH) and Auger recombination mechanisms in addition to optical generation which is already provided by Luminous. Mobility has been accounted for by using concentration dependent mobility and field dependent mobility models.FigureDevice Design: The array of trenches in the textured solar cell is approximated as rectangular holes in the 2D device design. The design shows surfaces textured with holes having 4 µm diameter and 1 and 5 µm gap values between the holes. The hole depth is 13.2 µm. The absorber thickness is limited to 100 µm instead of 500 µm to limit the number of mesh points (triangles) to be simulated, thus reducing the overall simulation time significantly. The simulation is then carried out by placing the beam for vertical incidence and changing the angle in steps of 5° up to 45° of incidence.FigureJsc reaches its maximum level at a beam incident angle of 15°and then falls off.The performance parameters of the cell were extracted from the simulation for two different pitch sizes, keeping the diameter and the hole depth constant. The plots of the various parameters versus the beam incident angle are shown in Limitations: The primary limitation in the simulations performed is the simulation of a cell with lesser absorber thickness (100 µm) instead of the dimension of the actual experimental device (500 µm). This results in a decreased value of the short circuit current and, hence, the efficiency of the simulated solar cell as compared to the experimental values. This limitation arises from the fact that as the beam angle increases, the number of reflections also increases. Thus, it becomes computationally prohibitive task in the case of large thicknesses, as the number of data points becomes huge. Although the simulated cell is thinner than the experimental cell, the trends are the same.2.1Figure For the samples with microholes, the efficiency first increases with the incidence angle and then decreases as also predicted by the simulation results. Although it is hard to obtain a perfect match between SILVACO simulation and experimental data we observe a convincing and qualitative agreement between them.Solar cells that were fabricated on Si surface with different distribution of periodic holes were analyzed at incidence angles varying in between 0° and 45°. A reference cell with standard pyramid texturing was also measured for comparison. As seen in It is seen that the microhole samples behave quite differently than pyramid textured samples and this feature have to be taken into the account for designing solar cells with hole textured surface. While a zero‐angle incidence is preferred in the case of standard pyramid textured solar cells, an incidence angle of around 15° is recommended for the microhole textured surfaces. The maximum efficiency value obtained from microhole samples displayed in Figure 3Angular dependence of the performance of a solar cell textured with microholes has been studied theoretically and experimentally and compared with the standard pyramid textured solar cells. Pyramid textured solar cells have exhibited highest performance at the normal incidence as expected while samples with microholes as the texturing features on the surface has shown a peak value at around 15° which is predicted by the theoretical simulation. Low performance of the microhole textured solar cell is related to the enhanced surface recombination at the surface whose area is increase due to the microhole formation, and also due to the poor metal contact formation.The authors declare no conflict of interest.


Kolumbo submarine volcano lies 7 km NE of Santorini caldera and its last eruption which occurred in 1650 AD, caused damage and casualties to the nearby islands. Here a simple model of a chamber, containing silicic magma underlain by a smaller quantity of mafic magma, is utilised in order to understand the chamber behaviour during the 1650 AD eruption. Results show that in order to reproduce the duration (83–281 days) and the dense rock equivalent volume ( Magma chambers residing in the upper crust . Geochemical analysis of the pumice samples revealed a rhyolitic composition with 17). This pre-eruption pressure corresponds to a depth of about 5 km and correlates well with the location of the low-velocity anomaly found in tomographic images13. As mentioned earlier, a significant finding of the petrographic analysis was the presence of mafic inclusions in the eruption products that suggest the injection of mafic magma in the chamber prior to the eruption. Despite the fact that Kolumbo represents a source of significant volcanic hazard for the nearby islands, there has not been any study to-date that has modeled the magma chamber evolution during the 1650 AD eruption and has drawn conclusions about the long-term eruptive potential of this volcano. The present work tries to fill this gap by addressing three interrelated topics: (1) what were the conditions inside the magma chamber during the 1650 AD eruption, (2) how these conditions can affect the frequency, duration and size of eruptions, and (3) what might be the long-term eruptive behaviour of Kolumbo’s magma chamber. In the next sections the physical parameters summarised above will be used in order to model the evolution of the magma chamber and obtain answers to these questions.Geological samples in the form of push/box cores and rocks from the crater wall of Kolumbo were obtained during several cruises in the area  tends to favor a conduit with a radius of about 30 m corresponding to an eruption rate of 15 who suggested that the deposits from 5 eruptions within a period of > 70 ka imply a recurrence interval of > 10 ka. However, before addressing the issue of eruption recurrence time, an estimate of the mass injection rate of mafic magma 22, or through the steady influx of melt. Seismological observations at Kolumbo tend to favour the latter possibility, since precise relative locations of microearthquakes cluster below the magma chamber at depths between 8–15 25. This seismicity has been recorded at Kolumbo during different time periods indicating that it is not a transient feature, but rather represents the response of brittle rocks to the steady migration of fluids in the magma chamber.A first assessment of the eruption frequency at Kolumbo has been made by Hübscher et al.26 have formulated a thermomechanical model for the eruption frequency of silicic magma chambers that reside in the upper crust and have also suggested scaling relationships that describe the long-term behaviour of such chambers. One of these relationships connects the number N of observed eruptions with the properties of the magma chamber and those of the upper crust such thatR with a percentage R, Degruyter and Huber15. These results also suggest that an eruption in the next few years is possible only if the chamber radius is smaller than 800 m. Such a chamber size is probably unrealistic for Kolumbo, since it would mean that its volume is equal to or smaller than the DRE volume of the 1650 AD eruption.The scatter plot in Fig. 26 defined three timescales that can help classify a magma chamber according to its eruption triggering mechanism and these are26: (i) triggering by second boiling when Another question that is related to the long-term behaviour of Kolumbo’s magma chamber has to do with the triggering mechanism of its eruptions. In their model, Degruyter and HuberR, 27. A small but steady influx of mafic melt is therefore necessary in order to keep the pressure in the chamber constantly rising.It is possible to use the randomised quantities  and K2 layer do not display any evidence of extensive mixing, but the mafic enclaves support magma mingling without any chemical hybridisationSimulations of the evolution of Kolumbo’s magma chamber during the 1650 AD eruption were performed by using a chamber model containing silicic magma that is underlain by a layer of mafic magma. Results showed that the conduit radius and the mass fraction of volatiles have the most significant effect on the eruption duration, while the eruption rate is also influenced by the initial overpressure in the chamber. On the other hand, the total volume erupted depends not only on the initial overpressure and mass fraction of volatiles of the silicic magma, but also on the quantity of mafic magma residing inside the chamber. Using as constraints historical and geological observations it is possible to conclude that the 1650 AD eruption likely involved an initial overpressure of about 10 MPa, a conduit radius of 40–30 m, and mafic magma that occupied up to 5% of the chamber volume.For a chamber radius consistent with the tomographic results at Kolumbo the average recharge time varies from 1.4 to 13.7 ka for mafic magma that occupies 1–15% of the chamber volume. This means that an eruption at Kolumbo in the foreseeable future is not likely, unless the magma chamber is much smaller  in order to model the evolution of Kolumbo’s magma chamber during its last eruption in 1650 AD, also with the aim of understanding its long-term eruption behaviour. The main conclusions of this study are as follows:27Q is the mass eruption rate, f represents the effective chamber volume divided by the effective compressibility for each magma type  form formr wh3434The bulk density of both the mafic and the silicic magma can be calculated by considering them as mixtures of melt, crystals and volatiles in which case the density is27G represents the cooling rate of the mafic magma. On the contrary, the temperature of the silicic magma is increasing as it receives more heat and the rate of change of its temperature is given byL is latent heat of crystallisation, The temperature of mafic magma is decreasing as heat is transferred to the silicic magma and the rate of change occurs according to the relationX can be calculated by the repeated use of the following equationX considered, while rand is a random number created by using Matlab’s random number generator. For the radius R of the magma chamber the chosen range was 200–1500 m, for the overpressure Randomised values of a quantity


Beta vulgaris (hr). LSP is based on the analysis of time‐resolved interference patterns. Speckle interference patterns of a biological object, known as biospeckles, are characterized by a dynamic behavior that is induced by physical and biological phenomena related to the object. Speckle contrast, a means of measuring the dynamic behavior of biospeckles, was used to assess the biospeckle activity. The biospeckle activity corresponds to processes modifying the object and correlates with the biomass growth. Furthermore, the stage of the cultures’ physiological development was assessed by speckle contrast due to the differentiation between active and low active behavior. This method is a new means of monitoring and evaluating the biomass growth of filamentous cultures in real time. As a potential tool to characterize hairy roots, LSP is non‐invasive, time‐saving, can be used online and stands out for its simple, low‐cost setup.Monitoring is indispensable for the optimization and simulation of biotechnological processes. Hairy roots  are producers of valuable relevant secondary metabolites. The genetically stable cultures are characterized by a rapid filamentous growth, making monitoring difficult with standard methods. This article focuses on the application of laser speckle photometry (LSP) as an innovative, non‐invasive method to characterize  B. vulgarisBeta vulgarishrhairy rootsLSPlaser speckle photometry1Plant cells and tissue culture generate a variety of valuable nutritionally, physiologically, and pharmaceutically relevant secondary metabolites. The in vitro processing of plant cells and tissue cultures is an investigative alternative to conventional techniques and a progressive means of producing a consistent quality and quantity of natural agents and additives under optimal conditions , 2, 3.Agrobacterium rhizogenes (A. rizogenes), which transfers the T‐DNA of the bacterial root‐inducing plasmid into the chromosomal plant DNA. Hairy roots are genetically stable and exhibit high growth rates and growth in the absence of plant growth regulators. The biomass growth of hairy roots is commonly classified as being in a lag phase, an exponential growth phase or a stationary phase. Hairy roots are characterized by a root‐like, filamentous structure ).Furthermore, the dry weight was analyzed in order to calculate the specific growth rate  is 1.65 ± 0.83 mg and is detected by an average biospeckle active area A (t = 3) of 1.02 × 104 px ± 2.16 × 103 px. After a short lag phase of 2 days, the curves of A (t) and mdw (t) rise simultaneously and exhibit exponential behavior. The exponential growth phase is identified between day 5 and day 14 of cultivation. Figure B. vulgaris  and low‐active areas Al [px]. The spatial distribution of biospeckle activity was labeled as active parts (green) and low‐active parts (red). The distribution of biospeckle activity for one representative culture is shown below.Furthermore, the values for AFigure As described above, the biospeckle activity is related to the biological and the physical dynamics of the object. The distribution of the biospeckle activity reflects the observation of biomass growth in Petri dishes. Branches in the center of the hairy root culture are older and their nutrient supply is limited due to confinement. Besides that, the growth of hairy roots is characterized by elongation of the tips. The dynamic behavior in root tips and in younger segments is indicated by the green areas. This visualization of the distribution of biospeckle activity in hairy root cultures is innovative and a progressive method to describe the state of the culture's development. The following Figure t, which is divided into a green part for active Aa and a red part for less active Al. As described above  can be detected and evaluated using LSP. LSP measurement takes only a few milliseconds, which means a saving of 48 hrs compared to the reference method. LSP is an simple optical, automated and rapid method with an affordable setup.The growth kinetics, growth phase and physiological behavior of The evaluation of biospeckle images is a means of gaining comprehensive temporal and spatial information about the objects of interest. Biospeckles provide an overall assessment of the sample and physiological states .4B. vulgaris. Biospeckle images were evaluated using speckle contrast analysis to measure the biospeckle activity. The biospeckle activity corresponds to processes modifying the object and correlates with the biomass growth. The increase in biospeckle activity occurs at the same time as the increase in biomass. The different growth phases  are monitored by observing the biospeckle activity during the cultivation. The maximum specific growth rates of all experiments calculated based on the change in biospeckle over the cultivation time are equivalent. The curve of the biospeckle activity as a function of the dry weight is linear until the end of exponential growth, then increases more slowly and ends with asymptotic behavior at the maximum dry weight. Besides this, after the biomass growth enters the stationary growth phase, the biospeckle activity decreases, which reflects the aging process of the cells. Furthermore, based on the distribution of the biospeckle activity, segments of the culture are divided into parts with active and low active behavior. The distribution of the biospeckle activity enables the cultures’ physical state to be evaluated, which is a completely new approach.LSP is a potential tool for monitoring biotechnological processes, which is indispensable for understanding, optimizing and simulating production processes. Plant cells and tissue cultures, e.g. hairy roots, produce a wide range of nutritionally, physiologically and pharmaceutically relevant secondary metabolites. Hairy roots are characterized by a filamentous structure, which makes common analytical methods difficult to apply , 3. Due B. vulgaris and to assess the cultures’ physical state or growth phase.As an innovative, non‐invasive, non‐destructive, rapid method that can be used online, LSP is a potential application to determine the growth of hairy root cultures of B. vulgaris (hr) in active and low‐active spatial distribution. For commercial use, the LSP technique and biospeckle method require an accurate interpretation of biospeckle phenomena [The speckle contrast as an index for biospeckle activity is currently classified for henomena . The modFurther investigations on the micro‐ and nanoscale are required to achieve this.Nevertheless, the LSP technique is a suitable and sensitive analytical method to determine the biomass growth of hairy roots in solid agar cultures. The time required to determine the biomass growth is significantly reduced and the optical setup is very simple and portable compared to existing methods. LSP is an approach that provides an overall assessment of the sample and physiological states.The authors are grateful for the financial support of this work by the European Regional Development Fund (ERDF) and the Free State of Saxony (project number: 100187245).The authors have declared no conflict of interest.