The model is formulated based on Bockris’s iron dissolution mechanism to describe the pH dependence of the corrosion rate. We also propose a simulation methodology to incorporate the thermodynamic database of the electrolyte solutions into the PF model. We show the applications of the proposed PF model for simulating two corrosion problems: general corrosion and pitting corrosion in pure iron immersed in an acid solution. The simulation results of general corrosion demonstrate that the incorporation of the anodic and cathodic current densities calculated by a Corrosion Analyzer software allows the PF model to simulate the migration of the corroded iron surface, the variation of ion concentrations in the electrolyte, and the electrostatic potential at various pH levels and temperatures. The simulation of the pitting corrosion indicates that the proposed PF model successfully captures the anisotropic propagation of a pit that is affected by the local pH of the electrolyte solution and the aggregation of Cl Laycock et al.1 simulated the growth of a single pit in a 304 stainless steel using the finite element (FE) method. A subsequent study revealed that the simulation results of the pitting corrosion on 304 and 316 stainless steels were in quantitative agreement with potentiodynamic experimental measurements2. Scheiner et al.3 developed a finite volume (FV) simulation methodology to simulate a two-dimensional evolving corrosion front in stable pitting corrosion. Onishi et al.4 developed a computational technique based on the FV method and the voxel method and analyzed time-dependent localized corrosion. Kota et al.5 proposed an FE-based simulation model to simulate the time evolutions of pit shape and the concentrations of ionic species in the electrolyte solution by solving the Nernst–Planck equation. They further applied a model to simulate stable pit growth in a polycrystalline microstructure in a SUS316 stainless steel. Brewick et al.6 also developed a FE-based simulation model for simulating the pit growth behavior that depends on crystallographic orientation of the polycrystalline microstructure in 316 L SS austenitic stainless steel. Yin et al.7 reported FE-based modeling to simulate the micro-galvanic corrosion of Al alloys where the Al dissolution rate depends on pH of the electrolyte solution. In order to consider the pH-dependent corrosion kinetics, the model incorporates the chemical-dependent current densities obtained by experiments as input data. Although the previous FE and FV-based models successfully simulated the corrosion processes, the models require explicit tracking of a moving interface between metals and electrolyte solutions to apply boundary conditions to the interface. Thus, the numerical simulations using the aforementioned models allow us to perform computationally expensive re-meshing and mesh refinement.It is necessary to understand the corrosion behavior of metals to develop metals with high corrosion resistance and to maintain structural metallic components. In order to predict the corrosion behavior, various computational models were developed to simulate the morphological evolution of material surface and the diffusion–migration of ions in electrolyte solutions9, grain growth11, and solid phase transformations16 in materials without the explicit tracking of moving grain boundaries and solid/liquid interfaces. As a pioneering study on the application of the PF model to simulate electrochemical reactions, Guyer et al.17 formulated the PF model that captures the formation of interfacial double layer in an electrochemical system. They also applied a PF model for electrodeposition and electrodissolution processes18. Subsequently, a few PF models that describe the stress evolution, the ionic diffusion and the interfacial migration in corroding materials were reported22. Stahle et al.19 constructed a PF model to simulate both pit growth and crack growth driven by stress corrosion cracking. Mai et al.20 proposed a PF model to simulate pitting corrosion in metallic materials and suggested that their model could simulate the evolutions of pits in the polycrystalline stainless steel and the ionic concentration in the solution. In a subsequent study, Mai et al.21 extended the PF model to simulate the stress corrosion cracking behavior in a polycrystalline microstructure and the galvanic corrosion processes22.Conversely, the phase-field (PF) method recently attracted attention as a powerful computational method that simulates a spatiotemporal evolution of material surface driven by electrochemical reactions. The PF method was originally developed as a computational methodology to simulate microstructure evolutions in solidificationet al.23, Kelly et al.24, and Keddam et al.26 already reveal that the dissolution rate of iron is strongly affected by the pH of electrolyte solutions. Additionally, the rate of the pitting corrosion significantly depends on the pH value of the electrolyte solution in the growing pit. Therefore, the development of a PF model that enables the simulation of the pH-dependent corrosion process is essential to predict the practical corrosion behaviors in a more quantitative manner.The above-mentioned PF models were developed to simulate various corrosion processes. However, to the best of the author’s knowledge, a PF model to simulate the corrosion behavior of iron and steels that depends on the pH of electrolyte solutions was not proposed in extant studies. Specifically, the experimental measurements by Bockris 27 is a commercial software developed by OLI Systems Inc. and provides thermodynamic data of electrolyte solutions, such as corrosion potential, anodic and cathodic current densities, with respect to various temperatures, electrostatic potentials, chemical compositions of alloys, and pH values of electrolyte solutions. Therefore, it is necessary to develop a simulation methodology to incorporate this type of thermodynamic data of electrolyte solutions into the PF model.Furthermore, with respect to various input parameters and data required to numerically simulate the corrosion, the thermodynamic data of electrolyte solutions constitutes extremely essential data to simulate the aqueous corrosion behavior quantitatively. For example, Corrosion Analyzer23 that is among the most widely accepted iron dissolution mechanisms that consider the pH-dependence of the iron dissolution. We also propose a methodology to incorporate the thermodynamic data of electrolyte solutions obtained by Corrosion Analyzer software into the PF model. In this study, we perform the numerical simulations of two typical corrosion problems by using the proposed PF model, namely general corrosion and pitting corrosion in a pure iron immersed in an acid solution. We examine these applications of the proposed PF model and demonstrate that the PF model captures the key features of the corrosion processes including the spatiotemporal migration of the corroded iron surface, evolutions of ionic concentrations, and local pH in the electrolyte solution, and the electrostatic potential during general and pitting corrosions.The present study proposes a new PF model to simulate the pH-dependent corrosion processes in ferrous metals. The PF model is formulated based on the Bockris mechanism+ irons, respectively. These reactions are expressed as follows:et al.24 reported, the iron dissolution in an acid solution depends on the pH of the electrolyte solution. One of the most well-known mechanisms of the pH-dependent iron dissolution is Bockris’s mechanism23. The electrochemical reactions proposed by Bockris et al. are expressed as follows:− ion in the solution to form the adsorbed intermediate Fe(OH)ads. Subsequently, as expressed in Eq.  occurs in the pit, and this causes the formation of H+ ion and the reduction of pH in the pit. The reduction of pH in the pit leads to the continuous growth of the pit. Therefore, the key features of the pitting corrosion that are needed to be simulated by the PF model denote the pH-dependent iron dissolution, the reduction of the pH, and the increase in Cl− ion concentration in the pit.In the pitting corrosion, the iron dissolution is localized at the iron electrode surface where the protective passive film is destroyed2+, H+, OH−, and Cl− ions.This section outlines the formulation of the PF model proposed in this study.  Although the PF model in conjunction with the Corrosion Analyzer software is applied to simulate general corrosion and pitting corrosion on a pure iron immersed in various aqueous solutions, the PF model is presented to simulate the corrosion processes in a pure iron in an acidic solution that contains Feξ , is defined as the local existence probability of the iron electrode, where r and t denote the coordinate and the time, respectively, ξ  corresponds to 0 in the electrolyte solution and 1 in the iron electrode, and ξ  changes continuously from 0 to 1 across a diffuse interface between the electrolyte solution and the electrode. The spatiotemporal evolution of the phase-field variable implicitly describes the migration of interface during the corrosion process. Additionally, ci   is defined as the concentration of ionic species i in the electrolyte solution. Hereafter,  is not expressed for a simple description.In order to describe the migration of the corroding iron surface, the phase-field variable, cFe2+, cH+, cOH−, cCl−} denotes the concentrations of Fe2+, H+, OH− and Cl− ions. Additionally, gdble(ξ) denotes the double-well potential energy density describing an energy barrier between the following two states: the electrolyte solution (ξ = 0) and the iron electrode (ξ = 1), and gdble(ξ) is given as follows:W denotes the barrier height of the double-well potential energy, and it is expressed as follows:b is a constant given by b = 2tanh−1(1 - 2λ), λ is a constant that defines the interfacial region as λ ≤ ξ ≤ 1 −λ, and λ = 0.1 is often employed. Furthermore, gact denotes the density of activation energy density and is expressed as follows:η is the overpotential of the electrochemical reaction given by Eq. . z denotes the number of electrons involved in the electrochemical reactions on the iron electrode surface, and F denotes Faraday’s constant. Additionally, h(ξ) is an interpolating function that is expressed as follows:The total free energy of the system is defined by the Ginzburg–Landau free energy functional as follows:n by Eq. . Althouggdble(ξ) and gact represents the local free energy density of the iron electrode and the electrolyte two-phase mixture, ggrad(∇ξ) denotes the gradient energy density that corresponds to the excess free energy due to the existence of interface between the iron electrode and the electrolyte solution, and gelec(ϕ) denotes the electrostatic energy density. The energy densities are given as follows:zi denotes the valance of ionic species i , and ϕ denotes the electrostatic potential. Furthermore, κ denotes the gradient energy coefficient and is given as follows:γ denotes the interfacial energy and δ denotes the thickness of diffuse interface.The sum of et al.30 and Chen et al.31, we obtain the following time evolution equation of the phase field variable:Lσ denotes the mobility of the interface between the iron electrode and the electrolyte that is moved by the gradient and the double-well potential energies. Lη denotes the mobility of the moving interface driven by the electrochemical reaction, and α denotes the transfer coefficient. Τhe activation overpotential, ηa, is given by ηa = E – Eeq where E denotes the applied potential of the electrode. Additionally, Eeq represents the equilibrium potential of the iron electrode in the electrochemical reactions given by Eqs . L(ξ) is used given that the iron dissolution occurs only in the interfacial region. Equation  by the electrochemical reactions on the iron surface. We assume that the variations of OH− and Cl− ion concentrations do not affect the corrosion rate compared to the pH of the electrolyte solution, and thus only the production of Fe2+ ion and the consumption of H+ ion are calculated.The time evolution of the ion concentration in the electrolyte is expressed by the Nernst–Planck equation that comprises of the diffusion of ions by the concentration gradient and the migration of charged ions by the electrostatic-field as follows:σ denotes the electric conductivity of the electrolyte and ρFe denotes the molar density of iron. Additionally, Δξ denotes the variation of the phase field variable representing the amount of the dissolved iron electrode. As assumed by Mai et al.21, the interfacial double layer is not considered in the study because the thickness of the interfacial double layer is in the order of nanometers. If we consider the double layer at the diffuse interface used in the PF model, it is necessary to use a very fine mesh with dimensions lower than a nanometer and a small time increment in the numerical simulation. Thus, it is impossible to perform the PF simulations of the practical corrosion processes in realistic time and length scales.We assume that the charge neutrality in the electrolyte solution, the distribution of the electrostatic potential in the electrolyte is given by solving the following Poisson-type equation:− concentration are controlled by adding HCl and NaCl, respectively. The details of the conditions used in the simulations are described in Methods.In this section, we present the applications of the proposed PF model to the simulations of the general corrosion and the pitting corrosion on a pure iron immersed in an acid solution. Initially, we show the results of one-dimensional (1D) simulation of the general corrosion. The 1D simulation of the general corrosion is used to examine whether the PF model can simulate the corrosion process that depends on the pH of the electrolyte solution and temperature. Subsequently, the three-dimensional (3D) simulation of the pitting corrosion is presented to demonstrate the applicability of the proposed PF model to a more complex corrosion process. In the simulations, we assume that the bulk pH of the electrolyte solution and the ClE = −250 mV vs. SHE. As an example, Fig. Ii, in Eq. , is defined as the local existence probability of the iron electrode: ξ  is 0 in the electrolyte solution and 1 in the iron electrode, the time evolution of the phase field variable shown in Fig. 2+ ion concentration increases in the solution with ϕ < 1. When the dissolution of the iron electrode proceeds with time, the Fe2+ ion in the bulk electrolyte increases. The profile of the Fe2+ ion concentration in the electrolyte is almost uniform because the diffusion of the Fe2+ ion in the electrolyte significantly exceeds the migration of the iron electrode surface.In order to examine a typical result of the general corrosion simulated by the proposed PF model, the simulation result for the initial pH of 2.5 at 60 °C (Case #2) is shown in Fig. + ion by the cathodic reaction in the interfacial region. It is observed that the amount of H+ consumption is time-independent and the variation of H+ ion concentration is extremely small, thereby resulting in a negligible change in the pH in the electrolyte solution. Therefore, the corrosion rate does not change during the general corrosion for 0.5 s. As shown in Fig. − and OH− ions are not considered in this study and the potential distribution in the electrolyte solution is uniform.The variation in the electrostatic potential is shown Fig. local ion concentrations for all species in the electrolyte solution.Although the corrosion rate appears as independent of the pH of the electrolyte solution, the simulation results in case #2 demonstrate that the proposed PF model captures the time-dependent migration of the iron electrode and the evolutions of the vPF, is validated by comparing it with the theoretical rate, vTh, which is calculated by Faraday’s law as follows:vPF indicates a good agreement with vTh for all pH values.The results of the PF simulations in different cases  are shown in Fig. In order to further demonstrate that the proposed PF model can capture the effect of temperature on the corrosion rate, Fig. Based on the simulation results of the general corrosion, we demonstrate that the PF model coupled with the thermodynamic data of the electrolyte as calculated by the Corrosion Analyzer software allows us to simulate the pH- and temperature-dependent general corrosion behavior of pure iron.2 is defined as the computational domain. Although the passive cover film is not explicitly considered, we assume that the passivated film covers the top surface of the iron sheet. The propagation of a single pit commencing from a circular breakdown of the protective passive film is simulated. It is assumed that the surface of the iron sheet in the pit is always depassivated during the pitting corrosion while the film on the iron surface is fully passivated when the cathodic reaction is maintained. The initial pH of the electrolyte and the temperature are assumed as 2.5 and 60 °C, respectively. The initial concentrations of H+ and OH− ions are calculated based on the pH and assumed as uniform in the whole electrolyte. The initial Cl− ion concentration is determined such that the iron sheet is immersed into the acid solution that contains Cl− ions equivalent to 1.0 M NaCl. The applied potential of the iron electrode E is assumed as equal to the corrosion potential and we use E = −420 mV irrespective of the pH of the electrolyte solution.The PF model used to simulate the general corrosion is subsequently applied to simulate the pitting corrosion on a pure iron sheet immersed in the acid solution. As illustrated in Fig. Lσ and Lη, were assumed as constant, because the pH in the electrolyte did not significantly change within the simulation. In contrast, the local pH inside the pit largely varies when the pit grows. Therefore, in order to describe the variation of the pitting rate by the local pH in the electrolyte, we evaluate the mobilities as the functions of the pH. (See Methods for the details of the pH-dependent mobilities used in this study).In the simulation of the general corrosion shown in the previous section, the mobilities, i.e., 2+ ion concentration increases in the solution within the pit due to the dissolution of the iron electrode. In the later stage, as shown in Fig. 2+ ion flows from the pit mouth to the bulk electrolyte. In contrast with the dissolution of iron, the reduction of H+ ion occurs inside the pit. Therefore, as shown in Fig. + ion migrates from the pit front to the mouth due to the electrostatic potential gradient shown in Fig. 32 reported that the pH decreases in the direction of the pit depth. However, the pH distribution in a pit strongly depends on the existence of passive film which covers the pit. In the case of the pit covered by the passive film, the transportation of hydrogen ion from the inside to the outside of the pit is prevented by the passive film, resulting that the pH increases in the depth direction. The phase-field model proposed in this paper also reveals that the pH in the pit decreases in the depth direction if no passive film covers the pit . The inhomogeneous pH distribution in the pit leads to anisotropic pit growth.Figure − ion concentration during the pit growth is shown in Fig. − ion are not considered in the study, the simulations reveal that the Cl− ion migrates to the bottom of the pit due to the gradient of the electrostatic potential and results in the aggregation of Cl− ions at the growing pit front. Based on the above-mentioned results, we demonstrate that the proposed PF model captures the key features of the pitting corrosion including the morphological change in the propagating pit, variations in the local pH inside the pit that strongly affects the anisotropic pit propagation, and ionic concentrations .Furthermore, the evolut1ion of Cl34, the pit depth increases linearly with the square root of time  activity coefficient is an important one to obtain the corrected pH values especially at high H+ concentrations 22)) and The new PF model was proposed to simulate the corrosion behavior of ferrous metals that strongly depends on the pH of the electrolyte solution. The present PF model was formulated based on the Bockris’s mechanism to implement the ability to describe the pH-dependence of the corrosion process. We also proposed a simulation methodology to incorporate the thermodynamic data of the electrolyte solution obtained by using the Corrosion Analyzer software with respect to the PF model. The proposed PF model was applied to the simulations of two corrosion problems, namely general corrosion and pitting corrosion in a pure iron immersed in the acid solution.− ion at the growing pit front. In future work, in order to simulate the corrosion behavior in iron more quantitatively, we need to consider not only the pH-dependent anodic current density, but also the pH-dependent other parameters: H+ activity coefficients, electric conductivity, and ion diffusion coefficients.The simulation results of the general corrosion demonstrated that the PF model successfully captured the migration of the corroding iron surface, local changes in ionic concentrations, and electrostatic potential in the electrolyte during the general corrosion. We investigated the effects of the pH of the bulk electrolyte and temperature on the corrosion rate. The results demonstrated that the PF model captured the increase in the corrosion rate with decreases in the pH value and increases in the temperature. Furthermore, the PF simulation of the pitting corrosion revealed that the proposed PF model successfully simulated the key features of the pitting corrosion including the morphological change in the growing pit that depends on the local pH in the pit, evolution of the electrostatic potential, and concentration of the ClΔx = 0.05 μm. The thickness of the diffuse interface is set as δ = 4Δx = 0.2 μm. The time increment is Δt = 2 × 10−7 s. The iron electrode with a thickness of 1.0 μm is placed at the left end of the domain, and other part is set as the aqueous solution. The initial concentrations of H+ and OH− ions in the electrolyte solution are set as constant at the values calculated from the pH. The initial concentration of Cl− ion is set based on the charge neutrality in the solution. The initial distributions of the ions are calculated by solving the Nernst–Planck equation with the fixed interface. The zero flux boundary condition is applied for the phase-field variable and the ion concentrations at both ends of the computational domain.The length of the computational domain is 2.0 μm. The domain is divided by 40 uniform finite difference grids. The spacing between the grids is IFe2+ and IH+ are calculated based on Eqs   IFe2+ anIn a numerical simulation, we calculate Eqs  and 17)17) by a The rate of the general corrosion is generally the order of a few mm/year. Thus, we artificially accelerate the rate of iron dissolution by decreasing the molar density of iron to check whether the proposed PF model captures the migration of the iron electrode surface and the evolution of the ionic concentrations in the solution. The acceleration corresponds to increases in the current density without using any unrealistic parameters as follows: Based on Faraday’s second law, the velocity of the migrating iron electrode surface is given as follows:ia,. However, ia also governs the production and the consumption of the ions at the electrode surface. Thus, instead of increasing ia, we accelerate the rate of iron dissolution by decreasing the molar density of iron, ρFe,. For example, if ρFe is decreased to ρFe* = ρFe/k, where k > 1, then the accelerated corrosion rate vd* is expressed as follows:The corrosion rate increases with increases in the anodic current density, k = 100.In this study, we use 3. The domain is divided by using 32 × 32 × 32 regular finite difference grids. We aim to simulate the growth of a pit whose size is a few micrometers. Therefore, the grid spacing is 1 μm. The thickness of the interface is set as δ = 3Δx = 3 μm. The thickness of the passive film is assumed as equal to a single finite difference grid. The zero Neumann boundary conditions at x = 32 μm are applied for Fe2+ and H+ ion concentration fields. With respect to other ion concentrations and the phase field variable, we applied the zero Neumann condition at x = 0 μm and the periodic boundary condition along the x and z directions. It should be noted that the acceleration of the corrosion rate used in the general corrosion simulation is not used in the simulation of the pitting corrosion.The size of the computational domain is set as 32 × 32 × 32 μmia0, Lσ0, and Lη0 denote the anodic current density and the mobilities at the initial pH of 2.5. Additionally, ia(pH) denotes the anodic current density at the corrosion potential and is assumed as dependent of the pH of the electrolyte solution. Furthermore, ia(pH) is calculated by Corrosion Analyzer software and approximated by the following polynomial functions:ia(pH) as obtained by the Corrosion Analyzer software and the approximated curve used in the pitting corrosion simulation.In the simulation of the pitting corrosion, we evaluate the pH-dependent mobilities by using the following equations:IFe2+ and IH+, used for Eq.   IFe2+ anSupplemental information


We present an efficient concept based on time varying and non reciprocal metamaterials to achieve an active control of the spoof plasmon (SP) propagation at sub-wavelength scale. An experimental demonstration of non-reciprocal guiding device based on split ring resonator is proposed as an application of this concept in the microwave regime. We show that this device is able to blue-shift the propagated SP waves and to achieve an active steering of these SPs at sub-wavelength scale by controlling the modulation frequency of the time varying metamaterial. This approach could be extended plainly to infrared and optical regimes by considering suitable technologies. SPs have been proposed for tailoring the light propagation at sub-wavelength scale thanks to the strong confinement of light at the metal-dielectric interface2. A series of applications based on SPs have been investigated theoretically and demonstrated experimentally, such as guiding and focusing devices3, photonic lenses4, wavelength demultiplexers5, integrated circuits6 and far-field sub-wavelength imaging7.Surface plasmons (SPs) are electromagnetic surface waves propagating along a metal-dielectric interface that can be observed in the visible wavelength regime9. This result has been obtained by cutting grooves on a scale much smaller than the wavelength to increase the penetration of the fields into the metal. These structures have been intensively investigated and has been utilized for many wave guiding and focusing applications16. Recently, second harmonics of spoof surface plasmon polaritons have been generated efficiently at microwave frequencies using a subwavelength-scale nonlinear active device17. This effect has been obtained over a broad frequency band by loading the nonlinear device to the intersection of two plasmonic waveguides with different corrugation depths.Spoof plasmons have been proposed to engineer SPs at THz and microwave frequencies18. Here, we show that the manipulation of the SP propagation can be achieved in a time varying metamaterial. This effect can be obtained in a planar subwavelength device composed of Split Ring Resonator (SRR) arrays that supports propagation of “magnetic” SPs at sub-wavelength scale. We show that a time gradient effect can be achieved by using an active resonator in the junction between three SRR based waveguides. A temporel modulation and an active manipulation of the SP propagation can be controlled by this time varying metamaterial. We show that the proposed structure is able to achieve an active switching device at a sub-wavelength scale.The dynamic control of metamaterial properties is attracting great interest in optics and microwaves in the last years. With the inception of time-gradient metasurfaces, it has been shown that Snells law can be modified to an even more universal form. By using this approach, new type of optical isolators has been developed by breaking Lorentz reciprocity18 given by:ϕ0 is the phase of the incident wave and ϕm the space-time varying phase shift in the metamaterial. This leads to a time variation on the frequency of the transmitted signal given by:f0 is the frequency of the incident wave. When we consider a normal incident wave and harmonic time variation of the metamaterial properties at a frequency fm, we obtain:It has been shown recently that a time varying metamaterial induces a phase shift on transmitted wavesThe above equation Eq.  describef0. Depending on the value of the modulation frequency fm, the SP coming from the first waveguide is then directed to the second (Port 2) or the the third waveguide (Port 3). This steering effect is provided by the bandpass properties of the meta-guides. In order to reduce the reflection of the remaining signal at the frequency f0, the first waveguide is extended beyond the junction, providing a supplementary output (Port 1) and collecting such remaining signal. The time varying metamaterial junction is able to carry out a switching between the second and the third output ports by controlling the modulation frequency fm. Thereby an active switcher for SP could be achieved by such device.Metamaterials supporting spoof plasmon wave propagation can be designed for a specific range from microwave to mid-infrared frequencies. Here, we propose to design a time varying non reciprocal subwavelength device based on such metamaterials in order to control the SP propagation. Figure 19. Such waveguides could be used here for the design of our switching device at infrared and even at optical frequencies. Electro-optical modulation could be then applied to achieve the time varying property of the metamaterial based junction. For instance, transparent conductive oxides (TCOs), specifically indium tin oxide (ITO), could be proposed to achieve this task due to their electrically-tunable permittivity in these regimes21.Now that we have discussed the basic principle and the physical mechanisms behind the control of the SP propagation, we move to the next stage concerning the design of the proposed time varying non reciprocal device. SPs offers a real opportunity to realize subwavelength components for modern electronics, especially in the infrared and optical regime. A wide variety of plasmonic and metamaterial based components have been recently proposed for these regimes. We have demonstrated recently the possibility of using defect SSP modes to realize bandpass metawaveguides22. Here, we propose to realize our subwavelength switching device by exploiting waveguides composed of SRR arrays. Such structures are very compact and can be realized conveniently by chemical, optical or electronic etching techniques23. Besides, grounded SRRs are able to provide painlessly bandpass filtering as it will be shown hereinafter.Here we propose to make an experimental demonstration of the SP propagation control in the microwave regime. Recently, the propagation of SP in an ultrathin metal strip consisting of connected split-ring resonators (SRRs) has been demonstrated at microwave frequenciesω0 is the resonant frequency of the SRR and κ is the mutual coupling between SRRs24. With a particular SRR orientation it is possible to obtain a bandpass waveguide supporting backward wave propagation (so that the phase and group velocities have opposite signs) at sub-wavelength scale.One dimensional array of SRR can be modeled by using a simple electric equivalent circuit and by applying Kirchoff’s laws. Typical dispersion relation of such arrays is given by:Figure GHz. This phenomenon relates the existence of the backward waves due to a negative mutual coupling as predicted by the previous dispersion relation.The dispersion diagram of the proposed meta-guide has been calculated by performing Eigen-mode simulations. The two first modes are presented in Fig. Three different arrays are then considered in order to provide three different bandpass filters for the realization of the proposed device. Time domain full wave electromagnetic simulations have been performed in order to optimize the transmission spectra of the designed meta-waveguides. Figure fm by using a sinusoidal signal at this frequency as bias voltage of the PIN diode. The switching between the two states at a frequency fm should provide the time gradient effect in the proposed device.In order to realize the time varying metamaterial, an active SRR based unit cell is designed. A PIN diode is inserted in a supplementary gap of the top printed SRR as illustrated in the inset of Fig. fm.The three meta-waveguides and the active SRR based unit cell have been realized and characterized in a semi-anechoic chamber. A spectrum analyser is used to obtain the magnitude transmission spectra of the waveguides. A software-defined radio (SDR) card is used to provide the bias voltage operating at the modulation frequency Figure f0 = 1 GHz is then injected in the main waveguide through a manufactured small loop antenna. Because this main waveguide supports SSP propagation at this frequency, the injected signal is able to reach the time-varying metamaterial junction. The modulation frequency fm of the junction is controlled by the bias voltage signal provided by the SDR card. This signal, set to 10 dBm, is able to achieve the switching of the PIN diode. A small loop probe is used to measure the spectra at the output ports.In order to validate experimentally the functioning of the time tarying and non-reciprocal switching device, the three waveguides are connected using an active metamaterial unit cell as illustrated in Fig. fm. In the first configuration, the modulation frequency is set to 220 MHz. In this case, the modulation of the injected signal in the junction leads to the generation of a supplementary signal at the frequency f2 = f0 + fm = 1.22 GHz. At this frequency, the propagation of SP is possible only in the second waveguide. Because, SSP propagation at the frequency f0 is forbidden in this waveguide, only the signal at the frequency f2 is able to reach port 2. However, both signals, at f0 and at f0 + fm, should vanish at port 3. Despite a low level pic at the frequency f0 depiceted in the port 2 spectrum, this scenario is confirmed by the presented measurement of Fig. f0 is driven by the main waveguide to port 1 as illustrated by Fig. f0 + fm and directed only to port 2 in this scenario.Figure fi = 0.98 GHz.One can notice here the existence of many pics not expected to appear in the measured spectra. These pics are essentially the harmonics of the modulation signal injected with a relatively high power. These harmonics are due to the non linear behavior of the PIN diode. We can also note here the appearance of one of the image frequencies at the frequency fm = 480 MHz. In this case, the frequency of the generated signal at the junction is f3 = f0 + fm = 1.48 GHz. The propagation of SP at this frequency is possible only in the third waveguide. Thereby, this signal is observed only at port 3 as illustrated by Fig. f0 is driven to port 1 as in the first configuration.Figure f0 is not possible. This phenomenon relates the non-reciprocal property of the proposed structure. However, sending back the shifted freuquency (f2 or f3) remains possible by considering the corresponding modulation frequency in each configuration.Because the selective bandpasses of the second and the third waveguides, transmitting SPs from port 2 or port 3 at the frequency With these two scenarios, the proposed structure is able to achieve a non reciprocal active switching of the injected signal by controlling the modulation frequency. This functioning is realized on a subwavelength device thanks to the small effective wavelengths provided by the SP dispersion relation. The obtained results demonstrate the possibility of controlling the propagation of SP by using time-varying metamaterials.In summary, we have proposed a time varying and non-reciprocal metamaterial to control the spoof plasmon propagation. A subwavelength device based on split ring resonators has been designed as an application of this concept. This non-reciprocal device is able to blue-shift the propagating SP waves and to provide an active steering of these SPs by controlling the modulation frequency the time varying metamaterial. By using such device, an active control of spoof plasmon propagation have been demonstrated experimentally in the microwave regime. Controlling the SP propagation as demonstrated could be realized by using different techniques that are able to impose an active variation of the effective parameters of metamaterials. This approach could be extended to infrared and optical regimes by considering suitable metamaterials and by employing specific techniques like the electro-optical modulation.Full wave commercial software (CST Studio) has been used to perform the numerical characterization. The calculations has been accomplished by the time domain solver of this software. Typical parameters of the Epoxy and the copper materials have been specified in the presented simulation results. The magnetic field cartographies have been calculated by the solver and post-processed in the final display.fm. The SDR has been controlled by a GNURADIO code where the maximum power has been set to 10 dBm. A manufactured small loop has been realized and used to excite the main waveguide. The measured signals have been collected using a commercial RF loop probe from AARONIA AG that was connected to the spectrum analyzer.The experimental characterization of the structures has been performed in a semi-anechoic chamber. An Agilent spectrum analyzer has been employed to measure the transmission spectra. The USRP B210 software-defined radio (SDR) card is used to provide the bias voltage operating at the modulation frequency


Evidence suggests that both the interaction of so-called Merkel cells and the epidermal stress distribution play an important role in the formation of fingerprint patterns during pregnancy. To model the formation of fingerprint patterns in a biologically meaningful way these patterns have to become stationary. For the creation of synthetic fingerprints it is also very desirable that rescaling the model parameters leads to rescaled distances between the stationary fingerprint ridges. Based on these observations, as well as the model introduced by Kücken and Champod we propose a new model for the formation of fingerprint patterns during pregnancy. In this anisotropic interaction model the interaction forces not only depend on the distance vector between the cells and the model parameters, but additionally on an underlying tensor field, representing a stress field. This dependence on the tensor field leads to complex, anisotropic patterns. We study the resulting stationary patterns both analytically and numerically. In particular, we show that fingerprint patterns can be modeled as stationary solutions by choosing the underlying tensor field appropriately. Large databases are required for developing, validating and comparing the performance of fingerprint indexing and identification algorithms. The goal of these algorithms is to search and find a fingerprint in a database (or providing the search result that the query fingerprint is not stored in that database). The database sizes for fingerprint identification can vary between several thousand fingerprints e.g. watchlists in border crossing scenarios or hundreds of millions of fingerprints in case of the national biometric ID programme of India.Clearly, fingerprint identification is of great importance in forensic science and is increasingly used in biometric applications. Unfortunately, collecting databases of real fingerprints for research purposes is usually very cost-intensive, requires time and effort, and in many countries, it is constrained by laws addressing important aspects such as data protection and privacy. Therefore, it is very desirable to avoid all these disadvantages by simulating large fingerprint databases on a computer.The creation of synthetic fingerprint images is of great interest to the community of biometric and forensic researchers, as well as practitioners. The SFinGe method  stress field, as one of the inputs of our interaction forces. Besides, the interaction force between two Merkel cells depends on the distance vector between these two cells. We model the coefficient functions of the interaction forces as damped harmonic oscillators, a well-established modeling assumption in cell biology. Besides, this choice reflects the exponential decay of the interaction over larger distances, implying that interactions over very large distances can be neglected, and reinforces an interplay between repulsive and attractive forces as the distance between two cells increases. This choice of the interaction forces is consistent with the general modeling assumption that interaction forces should be short-range repulsive to avoid collisions between cells, and attractive over larger distances to obtain cell accumulations. Note that a similar model is proposed in Burger et al.  and its Our class of models can be regarded as an biology-inspired adaptation of the Kücken–Champod model   \documeno et al. , Balaguéo et al. , b, 2014o et al. , Bertozzo et al. , 2012, vo et al. , Carrillo et al. , b, Fello et al. , 2011, Lo et al. , Raoul  w wF is ofthe form  and 6),,F is of the form  can be ring from  and simi and Sun  one can T is defined as  in inT intoentclass1pt{minimaIn the sequel, we consider the particle model  with forentclass1pt{minima such as . As in B such as  we consientclass1pt{minimaorces in , respectorces in , i.e. ths. In this section, we study the existence of steady states for the particle model ,,\documend| of the parallel vertical lines is consistent with the periodic boundary conditions.Note that the choice of the distance |n equidistant parallel vertical lines of n of parallel equidistant vertical lines, then the condition for steady states ,,n equidiorces in , implyinorces in . This immentclass2pt{minimn parallel, but not equidistant, vertical lines of Given mentclass2pt{minimle model  where thle model  and 6),,\documend| the line patterns in Proposition Even though parallel, equidistant lines form a steady state for any distance |mentclass2pt{minimMany non-constant tensor fields can locally be regarded as spatially homogeneous tensor fields. Note that by the assumptions in Sect. In this section we investigate how to simulate fingerprint patterns by extending the theoretical and the numerical results in Burger et al. . In partIn order to use the particle model  to simulFor simplicity we consider the origin \documentF is given by  as s(x) are defined to be unit vectors. In Fig. Given some real fingerprint data the aim is to construct the vector field mentclass2pt{minimConsidering the tensor field To quantify the distance to the steady state we consider the change of the positions mentclass2pt{minims of the tensor field and these bifurcations move towards the two neighboring lines over time. This behavior can be explained by attraction forces along the lines of largest stress over medium range distances, i.e. as soon as the distance between the particles along the lines of largest stress l is small enough they attract each other. In particular, the particles close to the bifurcation on the two neighboring lines are the first ones to ‘feel’ the attraction force along l and the two roughly parallel lines start merging close to the bifurcation. Hence, the single line on the other side of the bifurcation gets longer over time and the bifurcation moves towards the two parallel lines. While the two roughly parallel lines get shorter over time until they are finally completely merged, resulting in one single line. Since the movement of the particles is mainly along l there is a different particle at the bifurcation at each time step. While the particles on the line in the middle roughly remain at the same position apart from realigning along the lines of smallest stress s. This realignment along s is due to the additional number of particles which are aligned along one single line after the merging, as well as due to the repulsive forces along s spreading the particles to make use of the space along s and to avoid high particles densities after merging.In the simulations for spatially homogeneous tensor fields in Burger et al.  as well F in ,,F in (\deld T in  and the n forces  and 11)F in  is repulsive for large distances along l for the parameters in  ar arl for ficients . The foreters in . In Fig.entclass1pt{minimaentclass1pt{minimaIn this section, we consider interaction forces of the form  as befornt}fs in  and 28)\documentgiven by  and are given by  in compaeters in . Note theters in  are chosl,f¯s in , (28), rmentclass2pt{minims(x) is a unit vector and hence uniquely determined by its argument s(x) as s by the original fingerprint image in black in Fig. In Fig. mentclass2pt{minimIn conclusion, fingerprints with variable ridge distances can obtained as stationary solutions to our bio-inspired model. We consider harmonic oscillators as force coefficients, a well-established modeling approach in biology. Due to lack of experimental data the exact form of the interaction forces, including the parameter choices, cannot be validated with experiments. For this reason, the parameters are chosen such that certain observations are satisfied and the general model formulation of the model allows to consider a large class of models. As part of future work, the numerical results can be tested for realness. The distinction between real and synthetics could be based on Gottschlich and Huckemann  where hi


Objective. Neural interfaces and other implantable micro-devices that use polymer-encapsulated integrated circuits will only be allowed in medical devices when their lifetimes can be estimated from experimental data. An apparatus has been developed and tested that allows hundreds of insulated samples (interdigitated combs) to be aged under accelerated conditions of high temperature and voltage stress. Occasionally, aging is paused while the sample’s impedance is measured; the impedance spectrogram may show degradation as it progresses before failure. Approach. The design was based on practical considerations which are reviewed. A Solartron Modulab provides the frequency response analyser and the femtoammeter. The apparatus can accommodate batches of samples at several temperatures and with different aging voltage waveforms. It is important to understand features of the spectra that are not due to comb–comb leakage, but come from other places (for example substrate-solution leakage); some have been observed and investigated using SPICE. Main results. The design is described in detail and test results show that it is capable of making measurements over long periods, at least up to 67 °C. Despite the size of the apparatus, background capacitance is about 1 pF and comb–comb capacitances of about 30 pF can be measured down to 10 mHz, an impedance of about 100 GΩ. An important discovery was the advantage of grounding the bathing solution, primarily in that it raises the measurement ceiling. Observation and SPICE simulation shows that leakage from the substrate to the bathing solution can give phase lags  >90°, in contrast to comb–comb leakage which reduces phase lag to  <90°. Significance. The value of this paper is that it will facilitate research into the endurance of small implanted devices because, given a description of a proven apparatus, researchers can start building their own apparatus relatively quickly and with confidence. This paper presents one such design.Almost all electronic implants rely on polymer encapsulation to prevent leakage currents and corrosion. As implants have become smaller, electric field strengths have increased and yet far less research is done into the choice of materials and methods for encapsulation than into the electronic design. In addition to there being a wide choice of commercially-available materials, researchers would like to be able to test experimental formulations, compare methods of pre-encapsulation cleaning, and compare methods of encapsulation. How do these affect the lifetime of the device in simulated implant conditions? Since many samples of each type may be necessary to get statistically significant results, which can then be used to predict life interdigitated electrodes, a term used in the sensors literature where it was introduced by Hermens in 1984 [interdigitated combs because, as fabricated, they are capacitors with charge stored on the combs. Anderson et al started using IDCs to compare encapsulants and the effects of contamination due to inadequate cleaning before application of the uncured rubber; salt contamination caused liquid-filled voids to form which may lead to failure [The search for ways to provide satisfactory long-term insulation in implanted electrical devices is not new. PEK Donaldson published a paper in 1973 . IDCs arinsulating biomaterials were performed by Edell et al for many years, funded by the US NINDS Program. They published some remarkable results with a few samples which had lasted for up to 12 years at 37 °C and others for over three years at 90 °C [et al [et al [IDC tests of at 90 °C . More reC [et al  comparedet al provided little information about their apparatus, but described it applying constant voltages to stress (age) the samples. They made a 384-channel system, not multiplexed, with one current amplifier per sample [n [Much early work on insulation testing was done with constant applied voltages (DC). Edell ample [n , the numet al [The main advantage of EIS is that the impedance spectrum gives information about the sample being measured and how that changes during an endurance experiment. For example, Madani et al  fitted cet al . The preµm) on the top (circuit) side under a thicker conformal layer of silicone (~10 µm). Many questions need to be answered in order to inform these design choices and to justify implantation into patients:•Of several possible combinations of passivation layer and rubber, which have the longest time-to-failure (TTF)?•What is the aging acceleration due to temperature?•How does the TTF depend on the applied voltage waveform?•How does the TTF depend on the duty cycle?In the CANDO project, (Wellcome Trust: Grant 102037), a brain implant is being developed to treat epilepsy by optogenetic methods. The shafts of the optrodes are to be silicon and these support the LEDs, the recording electrodes, tracks and CMOS ASICs. The silicon will have one or two surface passivation layers , including photographs which show the apparatus and how it is arranged. Section The paper is arranged as follows: section 2.Figure In the following eight subsections of the paper, we have collected technical matters that were relevant to the design.2.1.The simplest model of a fault in an IDC is shown in figure fmin is the lowest frequency in the spectrum and Ctotal is the capacitance of the IDC and additional capacitance from the apparatus itself. For example, if, in figure Ctotal is 250 pF, with the void filled with pure water, the change in the slope of the magnitude spectrum will appear at a frequency of 7 mHz (when the reactance of 250 pF is 90 GΩ). The capacitive reactance is the ceiling of the measurement because higher shunt impedances cannot be detected. The higher this reactance, or the smaller the capacitance, the better.Where Requiring hundreds of samples seemed to have two consequences. First, the apparatus would have to be physically large, perhaps introducing significant extra capacitance in parallel with the IDC. Second, it was important that the time taken to measure each sample was kept short enough for all the samples to be measured at least every few weeks.•The susceptibility of measurements to interferences, given the extremely high impedances (up to 1 TΩ) and very small measurement currents (<1 pA);•The need for the apparatus to be physically large enough to accommodate all the samples, and yet not have such an inherent capacitance which would reduce its sensitivity by bringing down the measurement ceiling;•The samples must be held at elevated temperatures to accelerate aging;•The electronics of the apparatus must itself be protected from long-term exposure to moisture; and finally,•It must be practical to take measurements in experiments lasting years and with reliable data-collection.The design of the apparatus presented a large challenge, including:Assuming one can only afford one FRA, there is a conflict between apparatus sensitivity (impedance of detectable void) and the number of samples under test. Little has been published about apparatus for life-testing implanted devices by EIS. There are helpful general guidelines from Keithley  for high2.2.Modulab ECS, a potentiostat with an FRA and femtoammeter. The potentiostat of the ECS can be altered to a constant voltage source by connecting the reference electrode (RE) to the counter electrode (CE). This is what we have done in the work described below and consequently the input to the femtoammeter is described as the working electrode (WE) terminal, and the voltage source as the CE terminal.The apparatus utilises a Solartron auto mode: the latter has the advantage that resolution is maximised and random noise from the femtoammeter amplifier is minimised  but the2.4.2 in . The out2.3.µHz range but this is impractical; tests would take too long, and we have never worked below 1 mHz. The FRA generates the sinusoidal source voltage at CE. The experimenter sets the number of decades of frequency, the number of frequencies per decade, and the amplitude. Integration occurs over an integer number of cycles which may be fixed or be determined from the convergence of the average integrals  according to the set-up.For EIS, the applied frequency can be extended to the k and the spectrum is measured from fmax down to fmin, also supposing that at each frequency the FRA integrates over c cycles, then the total time will be (geometric series):EIS measurements are usually made at a predetermined number of frequencies per decade. If that number is k  =  5, which gives a good impression of the shape of spectra, and the frequency changes from 100 kHz to 0.01 Hz with c  =  3 cycles per frequency, then the total time is 811 s, close to 13½ min, of which 300 s, 5 min, is at the lowest frequency. Thus the total time is dominated by the lowest frequencies.For example, if 2.4.Another consideration is the effect of the substrate of the IDC. A formula was given by den Otter  for the A is the area of the IDC, s is the gap between fingers and a is the finger width plus the gap. For the IDC shown in figure A is 1 cm2, s is 60 µm and a is 80 µm. Suppose that the substrate is quartz  of 1 µm thickness, the calculated comb-substrate capacitance is 420 pF  with three reed relays connecting the Modulab to one of three IDCs, we found that if we used a mains power supply to provide the coil currents, the measurements were spoilt, whereas with a floating battery, very high impedances could be measured. The risk of electrical interference from other mains-connected sources also led us to choose optical fibres for the channel-select signals.We started by assuming that we would have some switches between the Shielding is a well-known method that is shown in figure Modulab is represented by the voltage source and the ammeter on the left. In figure Modulab. Shields are also sometimes called guards. In this paper we use the word screen to mean an electrostatic (earthed) sheet to limit an electric field in space. In passing, note that the impedance measured in EIS is the ratio of the source voltage to the ammeter current in figure Like Edell, we decided that each sample should be in a separate container so that they would be electrically isolated (but see sections module) from the backplane and connects it instead to the Modulab for half a day of impedance measurement. This is shown in figure aging voltage is applied to all seven IDCs in parallel or the impedances are measured one by one.If it takes about 20 min to measure the EIS of one sample down to 0.01 Hz section , then wi2.6.Figure Figure Below the bung, where the water vapour pressure is high, good insulation should be possible because the RTV silicone can remain bonded to the silicone tube for long periods in the presence of water, and so each conductor is intubated from the adaptor to the top of the bung. The RTV must be applied carefully so that there are no leakage paths between pins and also so that all metal surfaces (stainless steel and solder) are coated with RTV to prevent corrosion.The tops of the stainless steel pins are gold-plated so there are gold–gold contacts to reduce the risk of bad connections.2.7.Despite the fact that the o-ring that seals the bung in the tube is only silicone rubber, at 67 °C the rate of water loss from the tubes is negligible and has not need topping up after one year.To prevent leakage current not at the IDCs, in addition to shielding the WE conductor, it is also desirable to keep PCB surfaces dry. The PCBs are standard FR4 with solder-resist from ITEAD Intelligent Systems Co., not special high-resistivity types, and therefore it is important that water evaporating from the bath does not condense on the PCBs above. Figure In the tank, there are water-depth and water-temperature sensors; the water has to be topped up every few days. Above the baffle, in the airway, are temperature and relative humidity sensors, all connected to the GUI (see online supplementary data). Usually, the temperature above the baffle is higher than room temperature and the humidity is lower, presumably because of the warmth from the tank.The PBS in the tubes cannot be seen because of the large number of tubes and the baffle (see figure 2.8.•There is a socket on the left for the floating battery (~6.5 V) used during EIS measurements, to power the module without mains interference.•There are three fibre-optic receivers (red light) that carry a binary address that is used to select one of the seven channels (0 0 1 to 1 1 1).•The outputs of the decoder are connected to the gates of seven FETs which, when switched on, activate a reed relay and a parallel green LED showing which tube is selected.•On the right hand side is a 6-way DIN socket. If a plug is in the socket while the battery is connected, the buzzer sounds (to avoid making bad measurements).•One side of each transformer secondary (P1–P7) is connected to LO by a 10 kΩ resistor. These were added and are a significant modification. The advantages are described in section •The 5 V supply in the DIN connection causes the 1 MHz oscillator to excite the seven transformers in parallel which feed the depth electrodes. If the depth electrodes are in PBS, current flows in the primaries and also through the series red LEDs and the opto-isolators. This test is done for a few seconds every hour and not during sample impedance measurement.•The opto-isolator outputs are connected in series. If the PBS in any tube is low, the corresponding red LED will not light, and the impedance from the output LEVELS will be high. In this way, a row module with one or more depleted tube can be automatically detected and then the particular tube can be identified by looking at the LEDs.•The sockets for the tubes are mounted below the narrow horizontal PCB (see figure A in the supplementary data).•On the horizontal PCB, there are shield rings in the copper around the WEn connections on both surfaces.•The 2 mm stainless pins that project above the test tubes are inserted into gold-plated female connectors .•Hydro-X flux (Loctite HYDX 60EN), and the joints cleaned soon after soldering by rinsing in tap water.Beryllium–copper helical springs are used to accommodate poorly aligned pins without significant stress. These are soldered to the brass screws in the horizontal board and to the connectors with 60/40 Sn/Pb solder with water-soluble •The top and bottom of the horizontal PCB, plus the springs and solder joints to the connectors, are coated with adhesive silicone (Dow Corning 3140).•Each module has a grounded perforated metal screen on one side to reduce capacitive coupling between modules.•There are two variants of the module (A & B) because of the staggered rows of tubes in the tank figure .The complete schematic diagram for the measurement module is shown in figure After assembly and electrical testing, the lower half of the module is cleaned in a detergent solution followed by many rinses in water, finishing with 18 MΩ · cm deionised water, then blown dry in nitrogen and baked at 50 °C for 1 h before the silicone is applied to the horizontal board.The remainder of the hardware and the GUI are described in the online supplementary data.3.As explained in section In the experiment, the physical structure is much more complicated and yet we need to be able to model it. The IDCs are is in the test tube with its adaptor, wires and bung, all insulated, with PBS and depth electrodes (see figure C1–C6) between groups of nodes as shown in table We found these six capacitances for a typical sample figure  by makinThen the matrix equation  is solveTo get the values shown in table Notice that the capacitances between tracks on the IDC  are all small compared to the capacitances to the depth electrodes.CCD, CDS, CWD) inside a delta . If the depth electrodes are not connected to any other node, or it is justifiable to neglect this connection (see below) then there are only three nodes on each tube. In that case, only three capacitances are needed to represent it. Using the star-delta transformation, the three capacitances of the delta are given by:In the 4-node model of figure These capacitances are used in the simulations that are presented in section 4.4.1.Modulab while it is in air. Figure For baseline impedance measurements, we have a small Faraday cage  which allows the tube, containing the encapsulated IDC and PBS, to be connected to the 4.2.Three samples are compared in this test: (1) an encapsulated adaptor without any IDC attached to it, such that the wires from the bung go only to the adaptor; 2) an IDC sample for which the WE and CE tracks have been ablated by laser between the wire bond pads and the IDC itself  to be displayed. Both of these methods allow the interference that causes the erratic points to be inspected. A common cause of the outliers are glitches like that shown in figure As mentioned above, the These glitches may be infrequent so that, when one occurs at high applied frequencies, the FRA can continue integration until the average phase and quadrature currents have converged. But if the interval between glitches is similar to the period of the applied sine wave, then the average will not converge and an erratic point is produced. That is why the spectra become erratic at low frequencies.4.5.For some samples, we found that the phase at low frequencies did not do what we expected. If there is no leakage, the phase would be expected to be close to 90° of lag at low frequencies, but leakage would decrease the impedance magnitude and decrease the phase lag. What we saw was phase lags over 90°, as illustrated in figure We found that there are several advantages of connecting one of the depth electrodes to LO through a resistor. We chose a value of 10 kΩ, which is explained in section 4.6.grounding the depth electrodes. In addition to reducing the low-frequency phase lag, it has two other advantages. The first is that interference from the aging voltage generators is reduced because the PBS is acting as a screen, though not a very effective one, between the water in the bath and the DUT. The artefact when the aging voltage generator is applied to many other IDCs in the tank is shown in figure The connection of one depth electrode to LO is through a 10 kΩ resistor, which is very small compared to the impedances being measured, so we call this This useful effect will be explained in section The second, more important, advantage is shown in figure 5.5.1.Rf) current. The capacitor values come from table trans are between the primary and secondary of the transformers (measured as 28.4 pF). R1 and R2 are the actual resistors shown in figures P’stat) to the current in the femtoammeter (Rf). The Bode magnitude plot is shown in figure Figure •Ctrans from the depth electrodes nodes does not alter the impedance transfer function, so they can be removed. As there is then no extraneous connection to the D nodes (depth electrodes), the 4-node model of each tube can be changed to the 3-node model, i.e. from figures Disconnecting the capacitors •CΣΔWS, CΣΔWC and CΣΔCS. That gives the double delta model shown in figure Then lump together the three capacitances in tubes 1–6 and call them •CΣΔWC and R2 make little difference  so they can be removed.Observe that the two components •CΔWS in figure Observe that The full circuit of figure CΣ(ΔCS+ΔWS) is now (7  ×  CΔCS  +  6  ×  CΔWS)  =  1625 pF coming from all the IDCs. Figure ΔWC from the measured IDC)That leaves the much simplified circuit shown in figure R1  =  100 kΩ, Rf  =  10 kΩ and R1 alone, which is what it would be if the model were simply the 110 kΩ of R1  +  Rf in series with CΔWC.The transfer function of this network is5.2.We expect that the interesting changes that occur due to aging of the samples will be below the first cut-off frequency, about 1 kHz in figure Looking at figure CCS7 nor CWS7 have significant effect. Also there is no voltage drop across R2 at these frequencies, so it can be shorted, leaving the circuit of figure To see why this happens, we again looked for a simpler circuit model. Because we are here only interested in the low frequencies, we omitted the 6 tubes that are not being measured figure The phase in the voltages at D7 and therefore at W7 start to increase as the reactance of the three capacitors surpasses the resistance of the leakage resistance.Such sensitivity to very large leakage resistances is obviously important. Leakages from the depth electrode conductors to the shield conductors might occur in many places: at the bung, either above or below, in the measurement module, or between the depth electrodes and the shield track on the adaptor through the PBS.Because the effect on the impedance spectrum of such leakage is opposite to the effect of lateral leakage between the combs which we expect to see as the IDC deteriorates, there is a danger that these other leakages from the depth electrodes will confound the results.5.3.Figure Actually this is the same situation as in figure One of the advantages of adding these resistors is that leakage from the depth electrode conductors to the shield (LO) are less serious, because they are in consequence in parallel to the low-value resistors. This removes one effect that might otherwise confound the results.5.4.A remarkable point about figure CΣCS, CΣCD, CΣWD, CΣWS, CCS7 and CCD7 can all be combined to a capacitance CT of 6962 pF. CWD7 and CWS7 can be omitted because they are shunted by the low impedance of the femtoammeter. The capacitance in series with Rf is only CWC7 of 28 pF. The reduced circuit is shown in figure At low frequencies, the value of all the resistors is so low that they can reasonably be considered short-circuits. It follows that the capacitances CΔCS is found from equation  has been decreased from 233 to 28 pF. This is because the current cannot flow through the substrate via the capacitances between the substrate and the combs. By adding the 10 kΩ resistors, the PBS in the tubes is held at earth potential and that is capacitatively-coupled to the substrate through the thin layer of insulation so that it is also grounded. Consequently, current that flows from CE (i.e. voltage source) to the substrate flows to LO and not to the femtoammeter. Little current flows through the capacitance from the WE comb to LO because that capacitance (CWD7  +  CWS7) is shunted by the low impedance of the femtoammeter.Table The impedance spectra for the full model (figure 10 Ω asymptote at low frequencies. This shows that the addition of the 10 kΩ resistors between on depth electrode and LO in each tube raises the ceiling (background impedance), without altering the sensitivity to the leakage impedance that we are trying to measure.This raising of the measurement ceiling by a factor of 8 was an unexpected discovery and we wondered whether it might affect the sensitivity to the leakages between the combs that we want to detect on the IDCs. However, adding a leakage resistance to the full circuit model figure  shows th5.5.CWD and CCD, there is now an additional impedance Zleak between the substrate and the depth electrodes. The values shown in the figure were calculated using the formula for parallel plate capacitors for Cwsub and Ccsub and the known thickness of the thermal oxide. Zleak was measured for one sample  as a capacitance of 8.6 nF at 67 °C. This is much larger than one might expect from the substrate area and the average thickness of the silicone but we suppose that this is due to the very thin silicone covering of the corners of that substrate (which was dip-coated).Figure Zleak in figure Zleak is shown at the top of figure Figure It is often observed that the electrochemical cells that are leakages  actually have an impedance that has a constant-phase (CP) characteristic which means that the impedance magnitude falls at a rate that is slower than 1 decade/decade. To imitate such an effect, chosen resistor-capacitor pairs were put in series which gave the second curve in the top panel of figure 6.6.1.Modulab. In fact, with the shielding that we incorporated into the design, the capacitance when the reed relay connects the Modulab to the tube connectors, but without a tube present, is only about 4 pF compared to the unconnected cable from the Modulab of 0.8 pF. This shows the module adds very little capacitance to the cables.The development of the apparatus has been an expensive gamble. Practical-sized IDCs are only of the order of 100 pF which is quite a small capacitance. Had the background capacitance of the apparatus been large in comparison, the measurement ceiling would have been lower and consequently the additional multiplexing to accommodate all the samples would have spoilt the performance of the Modulab is not used to earth the screens. USB cables are all in the screened box under the bench but the Ethernet cable (Modulab to PC) is not (supplementary figure B). Despite the fact that the lab is in central London, at first floor level next to a busy road, and also close to the mains cables for the building, the Modulab can generally extract a satisfactory impedance spectrum despite the noise and interference seen at the femtoammeter output. We take precautions of switching off the room lights and discouraging entry to the lab  during measurements. The outstanding problem which still sometimes spoils the spectra at the lowest frequencies (and lowest currents) is the occasional current spike, usually of 10 s of pA, such as those shown in figure During development, we have spent some time reducing interference by arranging cable and screening circuits. All the mains cables originate from one connection to the supply, so there are no earth loops, and the LO output from the Modulab input sockets to the IDCs might spoil the measurements and water, perhaps condensed from vapour evaporating from the hot tank, might cause this leakage. The features of the design that meet this requirement (at least after a year with a tank at 67 °C) are the baffle to reduce vapour escaping from the hot water, ventilation above the baffle and the cleaning and encapsulation of the horizontal boards that support the tube connections (supplementary figure A). As measured day-to-day and shown on the GUI dashboard, the RH above the baffle is generally below 50%. Also important is the design inside the tube and under the bung, where RH  =  100%. Here we rely on the ability of some adhesive silicones to maintain effective glued joints in warm water: the silicone tubes pass right through the bungs so that adhesion to the bung (PEEK) is unnecessary.We saw management of water as critical to the design. Electrical leakage anywhere along the pathway from the 6.2.Through comparative tests with samples whereby the on-chip tracks between the IDC and the bonds pads were severed by laser ablation, we have shown that most of the measured capacitance is from the IDC itself table . HoweverThe purpose of the apparatus is to observe changes at the IDCs as a result of time, temperature and the aging voltage that stresses the sample. However, it is unavoidable, given the many samples and multiplexing, that from an electrical point of view, the apparatus is much more complicated than an IDC connected to the Modulab. Users of the apparatus need to be able to relate the measured impedance spectra to this complicated structure and we have shown that this can be done using SPICE models. As shown in section 6.3.We have described an apparatus that allows the Modulab to be used for monitoring many samples over a long period. It is a versatile arrangement as several different aging voltages can be applied, alternating, ‘DC’ or application-specific waveforms. The samples may be in baths set at the same or different temperatures. The effect of different bathing solutions can easily be tested because each sample is in a separate test tube. It is not limited to the IDCs shown in figure The apparatus can be used with small or large batches of samples. Our first experiment has been a materials comparison between three different passivation layers with two difference silicone encapsulants. With batches of 13, we see clear differences that will let us choose the best combination. In experiments in which the life of a device is to be predicted by accelerated aging at two or more elevated temperatures, batch sizes of 50 or more have been used (e.g. Merret ), and thIn the literature on the reliability of polymer-encapsulated integrated circuits, the reported failures are mostly corrosion of the aluminium  top metal not involving the deeper active structures which would be p- and n-MOS transistors in CMOS ICs. Nevertheless, one might want to investigate the stability of these transistors in long-term use. At least their leakage properties could be measured with this apparatus, if very large transistors were fabricated, and thus measure either their gate-substrate impedances or the drain-source impedances with the gates connected to the source. The Modulab can produce a wide variety of test voltages including alternating with superimposed bias, which might be appropriate for such transistor tests. At the time of writing we have only done insulation tests which are more fundamental to having a viable implanted device technology.6.4.Two recent papers should be mentioned on accelerated life testing polymer-encapsulated implants.n  =  4) so that they were continuously functioning  in an autoclave at 121 °C; none failed in a year [Since we started designing the apparatus described in this paper, the team at IIT in Chicago have published the results of an accelerated aging test on their Wireless Floating Microelectrode Array in which an integrated circuit is encapsulated in adhesive silicone rubber. They operated the devices .2.To validate the implant technology , tests will be done to look for early failures of Osenbach’s first two types. The experiment should be designed to examine the following questions. What fraction of samples fail in this way and over what period? Can these failures be predicted from changes in impedance? Could and should implants have a ‘burn-in’ (active soak test) to screen out incipient failures?3.For the wear-out process described by Osenbach, an experiment should investigate whether catastrophic failure can be predicted from prior changes in the impedance. If prediction is possible, the impedance data should be analysed to see what parameter(s) best predict time to failure. For the impedance-measurement to be really useful in accelerated testing, the time until actual failure should be substantially longer than the time until the impedance parameter(s) reach threshold.4.Some implants, or parts of implants, are not exposed to continuous bias, such as those driven by alternating voltages . It may With this in mind, the apparatus may be used in the following types of experiment.in vitro at elevated temperature(s) but such tests are likely to be expensive. The IDC tests might prove the technology before funding manufacture of actual implants and, having found the activation energy for the failure mode(s), may allow implants to be soak tested at only one temperature to confirm their lifetime performance.These tests are to be done on IDCs, not active implants, and they are intended to demonstrate the viability of the technology under realistic conditions of bias, duty cycle, etc. There is still an argument for testing actual implants 7.If the possible benefits from ICs & MEMs to implantable medical devices are to be realised, it must be shown that they can be sufficiently reliable. At present, the optimal materials and methods for protecting such devices are not established, and generally insufficient data is available to predict the lifetime in the body under the conditions of use . This paper describes an apparatus that can be used to compare materials and fabrication methods by applying accelerated aging tests with high temperature and voltage stress.Modulab with standard leads could conveniently be used with seven water baths which might contain 7  ×  96  =  672 IDC samples. If two modules are used per sample type, which is what we are doing, so 13 or 14 samples per type, then this capacity allows 49 types to be tested simultaneously. The semi-automatic arrangement means that the experiment needs a few minutes of attention each day that measurements are to be performed but this seems a good way to follow the changes that occur without being burdensome. After running the 67 °C bath for over a year, we are confident that at this temperature at least, there are no short-term problems from water harming the instrumentation.In our design, one A critical feature of such an apparatus is the measurement ceiling because this and the measurement time determine the smallest void that can be detected section . For IDCIn due course, we expect to report results from these experiments, both comparing different materials and investigating acceleration factors. We hope that this paper will encourage workers in other labs to set up dedicated accelerated life-testing apparatus and thus contribute valuable comparative data to our growing field.


Common applications include the quantification of the effect of different stresses , and the detection of trends and outliers. Pastas is a new and open source Python package for the analysis of hydrogeological time series. The objective of Pastas is twofold: to provide a scientific framework to develop and test new methods, and to provide a reliable ready‐to‐use software tool for groundwater practitioners. Transfer function noise modeling is applied using predefined response functions. For example, the head response to rainfall is simulated through the convolution of measured rainfall with a Gamma response function. Pastas models are created and analyzed through scripts, ensuring reproducibility and providing a transparent report of the entire modeling process. A Pastas model can be constructed in seven simple steps: import Pastas, read the time series, create a model, specify the stresses and the types of response functions, estimate the model parameters, visualize output, and analyze the results. These seven steps, including the corresponding Python code, are applied to investigate how rainfall and reference evaporation can explain measured heads in an observation well in Kingstown, Rhode Island, USA. The second example demonstrates the use of scripts to analyze a large number of observation wells in batch to estimate the extent of the drawdown caused by a well field in the Netherlands. Pastas is free and open source software available under the MIT‐license at  Article impact statement: Perform time series analysis of observed heads in a few lines of Python code with the open source software Pastas. Over the past few decades, time series analysis has become an accepted and frequently applied methodology in the field of groundwater hydrology. A particularly popular subdiscipline of time series analysis is transfer function noise (TFN) modeling, which attempts to translate one or more input series to an output series using a statistical model. TFN models can be used, for example, to decompose observed head time series into the contributions of the different hydrological stresses that cause the head fluctuations. Applications of TFN models include the estimation of the effect of interventions in the groundwater system (such as groundwater pumping and changes in surface water levels), the detection of trends, the improvement of data quality by identifying outliers and other suspicious data values, and the forecasting of heads.The first TFN models used in hydrogeology were autoregressive‐moving average models, which originate from econometrics  are the observed heads, hm(t) is the contribution of stress m to the head, d is the base elevation of the model, and r(t) are the residuals. Each model can have an arbitrary number of stresses (M) that contribute to the head; hydrological stresses include rainfall, evaporation, river levels, and groundwater extractions. The contribution of stress m to the head is computed through convolution:Sm is a time series of stress m, and θm is the impulse response function for stress m. A commonly used impulse response function is the scaled Gamma distribution  by one or more other observed time series. The basic model structure of a TFN model to simulate heads may be written as:t = 0, for example an instantaneous precipitation event of unit amount. The head response due to a uniform stress is called the step response, for example, a well turns on and starts pumping with a constant discharge. The head response due to a uniform stress for 1 day is called the one‐day block response and can be obtained through superposition of two step responses.There are three types of response functions. The impulse response is the head response due to an instantaneous stress event of unit magnitude at time t) due to a constant and unit stress starting at t = 0 may be obtained from the impulse response through integration:The step response Θ. Examples of the block response and step response functions for the scaled Gamma and the Hantush functions are shown in Figure The step response corresponding to the scaled Gamma impulse response function Equation  can be iR(t) from the rainfall P(t) and reference evaporation Er(t)  is substituted for Sm in Equation A stress can also be the combination of multiple stresses. An example of this is the linear model that is often used to simulate the net groundwater recharge α is the decay parameter, Δti is the timestep between observations at ti and ti − 1, and υ(ti) is the (approximate) white noise that is the result of a random process. The parameters of the TFN model may be estimated by optimizing an appropriate objective function, for example the sum of weighted squared noise . The residuals are modeled with a noise model (hence the name TFN modeling). An autoregressive model of order one is often used for time series with constant timesteps (Hipel and McLeod python_function) is used to refer to Python code. Information on how to download Pastas is given in the Acknowledgment section at the end of this paper.In this section, it is described how TFN models are implemented in the Pastas software. The typewriter font , keeps track of all the stresses and response functions that cause the head variations (stressmodels), the parameters in the model, the base level d (constant), the noise model, and basic model settings.The object‐oriented design of Pastas is relatively simple. Unified modeling language (UML) diagrams for the three most important classes are shown in Figure hm(t) are called stress models in Pastas. Each stress is stored separately in a StressModel object, together with the type of response function θm that is used to simulate the head response. Different response functions are available, including Exponential, Gamma, and Hantush. Each StressModel has a method to compute the contribution of that specific stress to the head variation, for example using convolution (Equation The model components to translate a stress (or two stresses in case of recharge) to a head contribution Equation , or anotRechargeModel and simulates the combined effect of rainfall and reference evaporation by using Equation RechargeModel object stores two stresses, the rainfall and (reference) evaporation time series, and one response function.A specific stress model class is available for the effect of groundwater recharge. This class is called Model class are (1) a method to estimate the parameters in the model, (2) a method to compute the simulated heads, (3) a method to compute the residuals, (4) a method to compute the noise, and (5) a basic method to plot the results. The method to compute the simulated heads is an implementation of Equation 1. The method loops through all M stress models and adds their contributions hm(t) to the head variations, then adds a base level d. The method to compute the residuals r(t) subtracts the simulated heads from the observed heads. The noise method applies Equation 7 to the residuals to compute the noise υ(t).The five main methods of the Series or DataFrame objects. Observed heads may be measured at arbitrary times. Pastas uses regularly spaced stress series in the simulations, and has functionality to convert irregular time series to regularly spaced series.Pastas builds heavily on Pandas, a data analysis package for Python that efficiently deals with time series data McKinney . All timImport the Pastas package.Series or DataFrame objects.Read the time series from files and store them as Pandas Model object and supply the observed head series.Create a StressModel objects by supplying the observed stress and specifying a response function, and add each StressModel object to the Model object.Create Model and compute fit statistics.Estimate the parameters of the Visualize output.Analyze residuals and noise.The basic workflow of a Pastas model is as follows:r2, root mean squared error [RMSE], and the Nash‐Sutcliffe coefficient), the Akaike and Bayesian information criteria, and a function to compute the autocorrelation of the residuals or the noise that deals with irregular time steps . The statistical capabilities include computation of various measures of the model fit , and the initial value for the scaling parameter of the response functions . While the heads are available on a daily basis, only biweekly observations are used in this example for demonstration purposes. The rainfall data is taken from the Global Summary of the Day dataset (GSOD) available from the National Climatic Data Center (NCDC) for Kingston station , located at 41.491°, −71.541°. The reference evaporation is estimated using Thornthwaite's method (Pereira and Pruitt The heads (site id 412918071321001) are obtained from the Groundwater Climate Response Network (CRN) of the USGS . After creation, the RechargeModel instance is added to the model.5Estimate model parameters. The model parameters are estimated by calling the solve method of the Model instance:tmax=‘2014’ is used, meaning that the model is calibrated on the available head observations up to 2014. A noise model and Scipy's least squares method (implemented as ps.LeastSquares) are used to estimate the parameters (default option). The solve method returns a report summarizing the model settings, the model fit, and the optimal parameter values and their estimated uncertainties, as shown in Figure In this example, 6Visualize model results. The results can be visualized with any of the predefined plotting methods, for example, the basic plot method. Here, the input argument tmax=‘2018’ is used to simulate and plot the heads up to 2018 .7Analyze residuals and noise. The residuals, noise, the autocorrelation of the noise, and a few other subplots to analyze the residuals and noise may be plotted with the following command:r2 value of 0.89 . The step response shows that the final level of 337 is reached after ∼300 d, which means that if it rains continuously 1 mm/d, the head in the observation well will eventually rise 337 mm after ∼300 d.The estimated step response is shown in the inset of Figure The observed head shows a peak in 2010 that is underestimated by the TFN model. The peak follows an extremely high rainfall event caused by the northeastern storms in New England in March of 2010. Apart from this extreme event, the observed heads can be simulated well with rainfall, reference evaporation (estimated from temperature data), and a simple TFN model.diagnostics plot that is created in Step 7, but are shown separately for this publication. The residuals can indicate if a stress is missing from the model, for example when a clear trend in the residuals is visible. As expected, the residuals are autocorrelated with extended periods where the modeled heads are higher or lower than the observed heads, while the noise shows much less autocorrelation. Based on the autocorrelation graph  autocorrelation in the noise, so that statistical inferences can be made about the model output.The residuals and noise are shown in Figure h Figure  it is coIn this second example, it is demonstrated how scripts can be used to analyze a large number of time series. Consider a pumping well field surrounded by a number of observations wells. The pumping wells are screened in the middle aquifer of a three‐aquifer system. The objective is to estimate the drawdown caused by the groundwater pumping in each observation well.3/d, or 12.7 · 106m3/year. The surrounding monitoring network consists of 44 observation wells at 23 locations (some are nested wells with screens at different depths) .Water extracted by the well field is used for drinking water supply in the southern part of the Netherlands. The well field consists of 21 pumping wells with screen depths between 110 and 170 m below surface level. A time series of the total extraction rate of all 21 pumping wells combined is available for modeling Figure . The ave) Figure . Daily r) Figure  is obtaiA Pastas model is created for each observation well using rainfall, reference evaporation, and pumping discharge as input series. Recharge is modeled with Equation As an example of one of the resulting models, consider well B49F0232 (Screen 5), located at ∼1500 m from the center of the well field. The model fit and the contributions of the net recharge and the pumping are shown in Figure A in Equation σA is the estimated standard deviation of parameter A. The contribution of pumping could not be determined with statistical significance for 9 of the 44 observation wells; these nine wells were not considered further.Using a Python script, 44 Pastas models are created (one for each observation well) that include the effect of pumping. After the parameters are estimated, it is checked whether the estimated steady contribution of the pumping response  estimates of the drawdown. The steady state drawdown for the average pumping rate over 2007‐2018 is computed for each observation well and plotted versus the distance of the observation well from the center of the well field in Figure Pastas is an open source Python package for the analysis of hydrogeological time series. The two objectives of Pastas are to provide a framework for developing and testing new modeling concepts and to provide ready‐to‐use software for practitioners. Pastas applies TFN modeling using predefined response functions. Multiple stresses  and corresponding response functions  may be added to a Pastas model. Models are created and analyzed through scripts, which makes the modeling process transparent and provides a full record of the entire modeling process. A Pastas model can be created in seven steps. These seven steps were illustrated (including corresponding Python code) in the first example, where the heads from an observation well in Kingstown, Rhode Island are analyzed. The second example demonstrated how a large number of time series models can be built in batch; the results were analyzed spatially to determine the drawdown caused by a well field.Pastas includes a large set of functionality, varying from a number of different response functions and the ability to select the calibration period to visualization and analysis of the results. All model concepts that have been implemented so far are for linear models: when the stress is doubled, so is the resulting head variation. This works very well for systems that behave more or less linearly, for example, relatively shallow (a few meters from the surface) wells in temperate climates. Pastas needs to be extended to be able to deal with other systems. For example, for deeper groundwater tables, the recharge is likely a nonlinear function of rainfall and evaporation  does not have any conflicts of interest.