Anomalous transport models for fluid classification: insights from an experimentally driven approach

Abstract

In recent years, research and development in nanoscale science and technology have grown significantly, with electrical transport playing a key role. A natural challenge for its description is to shed light on anomalous behaviours observed in a variety of low-dimensional systems. We use a synergistic combination of experimental and mathematical modelling to explore the transport properties of the electrical discharge observed within a micro-gap based sensor immersed in fluids with different insulating properties. Data from laboratory experiments are collected and used to inform and calibrate four mathematical models that comprise partial differential equations describing different kinds of transport, including anomalous diffusion: the Gaussian Model with Time Dependent Diffusion Coefficient, the Porous Medium Equation, the Kardar-Parisi-Zhang Equation and the Telegrapher Equation. Performance analysis of the models through data fitting reveals that the Gaussian Model with a Time-Dependent Diffusion Coefficient most effectively describes the observed phenomena. This model proves particularly valuable in characterizing the transport properties of electrical discharges when the micro-electrodes are immersed in a wide range of insulating as well as conductive fluids. Indeed, it can suitably reproduce a range of behaviours spanning from clogging to bursts, allowing accurate and quite general fluid classification. Finally, we apply the data-driven mathematical modeling approach to ethanol-water mixtures. The results show the model’s potential for accurate prediction, making it a promising method for analyzing and classifying fluids with unknown insulating properties.

Introduction

The measurement of electrical parameters is a common technique used for the characterization of matter properties. Several techniques have been developed for this purpose, and most of them are based on the measurement of current and voltage in different conditions [1]. The sensor described in the present study is intended to characterize fluids under relatively high fields, close to the breakdown field of the material. Insulating fluids are widely used in applications such as high voltage transformers, or more recently, in battery-operated equipments and devices. Electric cars are an example where insulating fluids can be used to cool down the battery pack. Liquid cooling fluids are also an efficient way to absorb and dissipate the heat generated by electronic components, as in computers and data centre hardware. The fluids do not conduct electricity and do not corrode the equipment, making it safe to submerge electrical equipment in them. However, contamination or degradation of the fluid can significantly reduce its insulating capability, thereby compromising both the safety and efficiency of the equipments. The problem of insulating fluid properties characterisation is generally solved by conductivity measurements or by dielectric strenght measurements. For very insulating fluids the conductivity measurement is generally not sufficient because the conducitivity is not a constant but is field dependent. For such kind of fluids the dielectric properties of insulating fluids are normally tested with specific equipments. In the present work we used as reference the instrument Megger OTF100AF Insulant fluids Tester. Such kind of testers are useful in laboratory environments but cannot be applied to field analysis during equipments operation. When high water percentage contamination is present such kind of instruments cannot be used. Moreover, methods based on the implementation of high voltages are expensive in terms of circuitry and electronic components and can generate other issues like electromagnetic noise and related electromagnetic compatibility issues. The proposed solution is quite new and generated a patent (WO2022254298 (2022-12-08): Sensor Device for Monitoring the Dielectric Strength of a Dielectric Fluid, in particular a Fluid for the Thermal Conditioning of a Battery), where the concept of obtaining high electric fields at low voltage by micro-gaps is used. Indeed, while discharge phenomena in micro-gaps have been studied in the literature primarily to understand failure mechanisms in microdevices such as electrostatic actuators (see, for example, [2]), they have not yet been applied to the characterization of insulating materials.

The idea underlying the new sensor is that reducing the distance between electrodes it is possible to obtain very high fields, even at low voltage. Scaling of the electrodes distance allows the measurement of current flow at low voltage. As we will demonstrate, the charge transport characteristics at microscale and with an high electric field, but below the breakdown voltage, is predictive of the insulating properties of the fluid in standard laboratory conditions with reference equipment.

The flow of the electric charge in the media between electrodes can follow different regimes, not only reducible to ohmic behavior but including different types of anomalous diffusion. In the first scenario, the mean square displacement of the transported entity is characterized by a time-linear dependence, , which is typical of a Markovian process, and the overall dynamics of the system is described by the diffusion equation. In contrast, the process exhibits anomalous transport when it is characterized by a different time dependence for the mean square displacement, i.e. , providing evidence of the non-Markovian nature of the process, [3, 4]. In fact, anomalous electric response is found in several systems, such as fractal electrodes [5], nanostructured iridium oxide [6], and water [7, 8]. Several approaches have been proposed to mathematically describe these anomalous responses. Microscopic models focus on the evolution of systems made of a certain number of particles, and include molecular dynamics techniques [9, 10], continuous-time random walk, Lévy flights and fractional brownian motion. While microscopic models gain important insight into the collective behavior starting from the dynamics of the constituting particles and their statistical properties, but at quite high and often prohibitive computational costs, macroscopic models offer a simpler and computationally much cheaper framework for understanding how these behaviors scale up to affect transport processes at the macroscopic level. Specifically, macroscopic models describe the time evolution of particle densities, including different nonlinear diffusion models and fractional diffusion equations. Fundamental reviews can be found in [3].

In this study, we focus on macroscopic models, specifically four partial differential equations (PDEs): (i) a generalization of the Gaussian model with time-dependent diffusion coefficient, (ii) the Porous Medium Equation, (iii) the Kardar-Parisi-Zhang Equation, and (iv) the Telegrapher Equation. In order to inform the mathematical models, specific experiments are carried out to explore the transport properties of the elecrical discharge observed within the micro-gap based sensor immersed in fluids with different insulator properties. Even if the sensor was initially designed to measure the properties of insulating fluids, it has been then used also with different kinds of conducting fluids. Through data fitting, each fluid is then characterized by the optimal parameters which reveal the transport properties of the process and represent a signature of the material where the discharge at the microscale is measured. We find that positioning different materials within a parameter space is a simple yet powerful tool for estimating the insulating properties of fluids, with results in good agreement with laboratory test procedures. The proposed methodology thus supports the development of a new class of sensors, where a relatively simple hardware can supply deep understanding of the transport phenomena. These sensors will be miniaturised, portable, with low power consumption and cheap. These characteristics make them ideally suited for in-situ applications, where real-time monitoring of insulating fluid conditions is essential. The rest of the paper is organized as follows. In Sect. 2, the experimental setup is described with the characteristic of the micro-gap sensor used to collect experimental data. Section 3 introduces the continuous models for anomalous transport mentioned above. In Sect. 4, we perform the models calibration based on experimental data which allows us to rank the models in terms of fitting performance and identify the TdDC model as the most effective model in describing the observed phenomenon. We then apply the fitting procedure to fluid classification, using both fluids with well known insulating and conductive properties as well as ethanol-water mixture. We conclude with a discussion and suggestions for future investigation.

Motivating experiment: nanotechnology-based sensors

The core of the experimental setup is the micro-gap: it is formed by a couple of microelectrodes fabricated by photolithography at a distance of few micrometers. Different distances, materials and versions have been tested but a typical micro-gap is depicted in Fig. 1 (see also Fig. 2A). The minimum distance between the electrodes is of 1.5 microns. The micro-gap is wired to a switch that can connect the micro-gap to a 100 nF capacitor (see n. 1 in Fig. 1). The capacitor is connected by a second switch (n. 2 in Fig. 1) to a DC power supply (model EA-PS 5040-40 A). When the capacitor is under power supply charge, it is disconnected from the micro-gap (switch n. 2 closed and n. 1 open). When the contact between the power supply and the capacitor is open, the capacitor can be connected to the micro-gap. The voltage on the micro-gap is measured by an oscilloscope (model RS PRO IDS-1104B). We refer to Sect. 6 for additional details on the experimental setup, sample preparation, and data reproducibility. If the micro-gap is immersed in a virtually perfect insulator, the discharge time of the capacitor is virtually infinite. In real media a current can flow between the microelectrodes even at finite times. The current flow in the micro-gap can then be used to characterize the fluid, as it will be clear in the next sessions.

Fig. 1.

Functioning Block Diagram. A capacitor is fully charged from power supply (switch 1 open, 2 closed), then it is connected to the Micro-Gap that allows electrical discharge (switch 1 closed and 2 open). The sensor allows to directly detect a possible decay of the dielectric strength of an insulating fluid

Fig. 2.

A Micro-gap made of a metal thin film, typically Cr 100 nm thick, deposited on an insulating layer (SiO or AlO) with electric contacts (Al or Au, 150 nm thick). B Detail of the part of the device where the distance between electrodes is minimum. The spatial domain is set to represent the micro-gap extension

Anomalous diffusion models for the discharge between micro-electrodes

As mentioned in Sect. 1, transport phenomena at the nano-scale are commonly expected to be anomalous, because the microscopic conditions for the establishment of local thermodynamic equilibrium cannot be guaranteed. This is the condition that allows matter to be treated as a continuum, so that balance equations for what flows in and what flows out of a given volume element, be it mass, charge, momentum, energy etc. can be written as partial differntial equations. For instance, such equations include Ohm’s law of electric conduction, Fick’s law of diffusion and Fourier’s law of heat conduction, that arise in the linear regime in which driving thermodynamic forces are proportional to the currents they induce [11–14]. If local equilibrium holds, linear or non-linear laws of continuum apply, but if it is violated, different approaches may be required. This can be understood as follows.

Because matter is made of particles that move erratically in space, treating it as a continuum requires three widely separated space and time scales: the microscopic, the mesoscopic and the macroscopic scales

1

within which correspondingly different phenomenologies occur. We get an idea of this thinking of a cloud and the water droplets that make it. The cloud looks like a continuum when observed from the ground, i.e. on a scale in which the droplets do not appear. Here, is the characteristic microscpic distance; the corresponding charcteristic time; L is the typical system size at the level of observations; and t is the time required by a macroscopic measurement tool to report a reading. The distance must be large enough for a cube of volume to contain a sufficiently large number N of properly interacting particles, that fluctuations of this number due to particles entering and exiting the box be negligible. This is required for the properties concerning that volume element to be sufficiently stable that it makes sense to measure them with macroscopic tools and to consider their variations as given by derivatives. Interactions, randomizing the motion, contribute to this stability. The randomization of motion also yields, for sufficiently large N, a uniform distribution of mass, charge etc. so, for small boxes centered at a postion x, they can be represented by single numbers , c(x) etc. It is also required that the range of interactions be short compared to , so that the quantities of interest appear as local quantities: the influence of this box on the properties of matter in other volume elements is negligible. The time must be long enough for the density and any other observable to have become homogeneous in . At the same time, must be short enough to be perceived as an instant at the macroscopic level. Also, for matter to appear as a continuum, it is required that the mesoscopic cells be so small that they look like a single point on the scale of L. All these conditions can be satisfied if, in addition, driving forces are small [11, 13]. When this is achieved, the mass in constitutes a small “isolated” thermodynamic system, and the collection of all such thermodynamic systems can be treated as a continuum, with good differentability properties. Transport phenomena arise as small effects compared to the mass and energy stored in the mesoscopic cells. For instance, the energy constitutes the huge reservoir known as internal energy.

In the case of transport of electric charges, a simple example is afforded by electrons in copper at 273 K. In this case seconds [15]. The Fermi level energy is eV, with an effective mass . Consequently, because kg; m/s, we have the Fermi level electron speed m/s and the microscopic characteristic length m. Then, in a cube of side m one expects collisions per collision time , and in a time interval of seconds, relaxation to local equilibrium is achieved in that cube. This means that can be taken to be of the order of seconds in copper at 273 K. Therefore, times of order of sec and volumes larger than m are expected to make local thermodynamic equilibrium applicable, if electric potentials are not too large.

On the other hand, processes occurring within distances of the order or smaller than m or over times shorter than seconds, under even moderate potentials, could violate Ohm’s law, to some degree. This fact may even be used to devise particularly sensitive instruments, see e.g. [16]. However, when the conditions for local thermodynamic equilibrium are violated, it is hard to tell what a phenomenon could take place, because finite size [17] and contrasting effects [18] make it highly dependent on all variables and parameters describing the system at hand. Thermodynamics may still apply, typically being not robust under perturbations, or may not apply, in quite unpredictable fashions [16, 19, 20].

Our experiments have some similarity with electrical breakdown experiments and lightnings, that violate Ohm’s law [21], as they are triggered by very large potentials that drastically modify the medium in which they take place, e.g. ionizing it. At the same time, the materials we use are quite different from the atmosphere, and concern totally different scales. Therefore, it is conceivable that our experiments fall in the realm in which local thermodynamic equilibrium is not guaranteed, but which mechanisms are really at work, needs to be clarified. This is one of the purposes of our investigation.

We remark that, to the best of our knowledge, there are no previous studies that apply similar anomalous diffusion models to experimental data on electrical discharge within micro-gaps. To fuflfill this goal, we have considered various models that give rise to anomalous transport, as well as the Telegrapher’s equation, that is used in the study of lightnings [21]. Each of these models is potentially suitable–based on different physical considerations–for describing the observed phenomenon. These equations have to be understood as phenomenological models, that require experimental tests to decide which is most suitable for our experiments.

The experimental setting is modeled in terms of the voltage u(x, t) at position x in the one-dimensional domain representing the micro-gap extension (see Fig. 2), for all times t in a given interval [0, T]. We fix proper initial and boundary conditions, that will be then coupled with different evolution laws describing anomalous transport within the micro-gap. Specifically, we set an unbalanced initial voltage configuration, represented by a reflecting boundary condition at and absorbing boundary condition at . In mathematical terms, we write:

2

where is the initial charge on the left electrode.

We then consider different evolution laws such that the mean square displacement of charges, is asymptotically proportional to , with .

Gaussian model with Time Dependent Diffusion Coefficient

The Gaussian model with Time Dependent Diffusion Coefficient (TdDC) derives from a generalization of the Diffusion equation which includes a general spreading of the density, i.e. anomalous diffusive transport (for a detailed derivation, see [22]). It reads as

3

Specifically, it introduces a temporal dependence of the diffusion coefficient which describes normal diffusion for and anomalous diffusion for , see the numerical solution in Fig. 3A. In the second case, implies clogging, i.e. slowing down in time of the transport, that may even stop altogether; while means that this rate rapidly increases, allowing a burst or a sort of micro-lightning. Analogously to the fractional Brownian motion [23], the time dependent diffusion coefficient allows the properties of transport to change in time. In particular, implies impediment in transport at short times, and rapid transport at later times. This is analogous to the electric breakdowns in which the dielectric properties of the medium have to be altered before the discharge can take place, and when it takes place it is fast and accelerates till the end. When , transport is initially fast, but it develops in time a kind of clogging phenomenon. Equation (3) is thus a promising single model, suitable to describe quite diverse situations with only two parameters.

Fig. 3.

Time evolution of models solution. A TdDC model, for different values of and . B PME model, for different values of and . C KPZ model, for different values of k and . D TE model, for different values of and . Parameters in the initial condition are set as ,

Porous Medium Equation

The Porous Medium Equation (PME) describes density-dependent diffusivity in porous media

4

which has been proved to be related with anomalous transport both from theoretical [24, 25] and experimental [26] studies. Moreover, simple algebra provides an explicit relation between the exponent appearing in the PME and the anomalous transport feature of its solution, i.e. (for explicit derivation, see [22]). Equation (4) thus describes normal diffusion for and anomalous diffusion for , see the solution profiles in Fig. 3B. Equation (4) has properties analogous to those of Eq. (3), with the difference that the change in efficiency of transport, characterized by the effective diffusion coefficient , takes place when the density of the transported quantity changes, rather as time passes. In a sense, this provides a mechanism for different kinds of transport, again with two parameters to fit, that has proven useful even in exotic phenomena such as transient osmosis [19]. This makes it another interesting model to account for the results of our experiments.

Kardar-Parisi-Zhang Equation

The Kardar-Parisi-Zhang (KPZ) equation was first introduced to recapitulate universal aspects of growing interfaces [27], and later connected to anomalous transport phenomena [28]. It reads as

5

where k and D are positive constants, and is white Gaussian noise with average and instantaneous decay of correlations: . It has been used to describe different physical phenomena such as turbulent liquid crystals [29], crystal growth on a thin film [30], bacteria colony growth [31, 32] and burning fronts [33]. We here neglect the noise term, i.e. we set , because our tests revealed that noise is minimal or totally absent in our experiments. Figure 3C show the effect of varying the parameter k on the resulting solution. The non-linearity of the KPZ equation accounts for intricate evolutions, that constitute a large array of situations, including critical phenomena, and scaling properties. These characteristics make it fundamental in understanding non-equilibrium physics, hence an important tool to test also in our experiments.

Telegrapher Model The Telegrapher model describes a finite-velocity diffusion process. It has been derived in the context of electrodynamic theory [34], and later proposed as a model for the transport in turbulent diffusion [35] as well as for the lightning processes [21]. It is formulated as

6

where and denote the diffusion and damping coefficients. For more details on derivation and application of the telegrapher’s equation, we refer to the literature [36, 37].

The main interest for Eq. (6) here is its ability to describe various facets of lightnings. In fact, it describes how voltage and current waves travel along a transmission line, and a lightning is an electromagnetic wave propagating through the atmosphere. It describes reflections and transmissions of waves at discontinuities in transmission lines, that can be observed in lightning waves at different atmospheric layers or ground interfaces. It allows for distributed resistance, inductance, capacitance, and conductance along the transmission line. It can also describe the formation of complex wave patterns. All these characteristics, although could quantitatively differ from our electric discharges, can be found in our experiments. Therefore, this is another model worth trying.

Figures 3D show how variations in the damping coefficient affect the resulting solution.

Results

In this section, we inform and calibrate the PDE models introduced above to shed light on experimental time series data of voltage discharges observed between the sensor micro-electrodes.

Models calibration based on experimental data

Experimental time series data on voltage discharge between the sensor micro-electrodes are collected and used to investigate which model better describes the physical phenomenon. We refer the reader to Sect. 6 for details on data collection and reproducibility. We assume that the voltage measurement is represented by the model’s solution at the left boundary of the domain, u(0, t). The other electrode is grounded, . Model calibration is then performed using a classical Bayesian method, leveraging the built-in MATLAB routine lsqcurvefit, which is based on the least-squares method. Specifically, given the experimental observations , for , the optimal parameter set is obtained by minimising the sum of squared residuals

7

where is the model solution obtained for the parameter set p. Figures 4 and 5 compare the time evolution of u(0, t) for different model parameter values (panels A-C) and the fitted solutions obtained by adapting the models to the voltage data observed when the micro-electrodes are immersed in two illustrative fluids with opposite conductive properties, i.e., air and tap water (panels B-D). Figures 4B, 4D, 5B and 5D show that all the PDE models considered here can qualitatively describe the electrical discharge.

Fig. 4.

A–C TdDC and KPZ solutions at . Models parameters are set as , , . B–D fitted solutions obtained through model calibration to experimental data

Fig. 5.

A–C PME and TE solutions at . Models parameters are set as: , , (PME); , , (TE). B–D Fitted solutions obtained through model calibration to experimental data

To determine which model quantitatively best replicates the phenomenon, we evaluate the squared 2-norm of the residual obtained for the optimal parameter set, i.e. R defined in Eq. (7), to quantify the goodness of the fit for the four PDE models. The smallest R identifies the best performance. For this purpose, we use measurements of electrical discharge observed when a 12 V electrical voltage is applied to micro-electrodes immersed in fluids with varying conductive properties. Specifically, both insulating fluids, i.e. air, 3 M Novec 7100, OPTEON SF33 (HFO-1336mzz-Z) (both insulating fluids for refrigeration and cleaning, featuring high volatility), isopropyl alcohol, SHELL DIALA S4 ZX-I (insulating oil for power transformers), as well as conductive fluids, including tap water, deionized water, and ultrapure water, are used. The mean and standard deviation of R, evaluated over a set of ten experimental replicates for each fluid, are shown in Fig. 6. Specifically, for each of the 10 replicates, R is evaluated as the minimum value obtained by the fitting procedure upon the choice between three different values of the steepness n of the initial condition, i.e. . However, the option is excluded for the TE due to numerical instability issues. Overall, the best fit performance is obtained by the Gaussian model with Time dependent Diffusion Coefficient (TdDC). Moreover, Fig. 7 shows that fixing does not significantly affect the performance of the fitting procedure. Because we also need a steep initial profile, to represent the situation before the discharge, we will adopt this choice, and the TdDC model, in the following sections.

Fig. 6.

Performance comparison of the fitting procedure applied to the PDE models, with the steepness parameter n of the initial condition chosen among three values (, and ). For each fluid, the mean and standard deviation of the squared 2-norm of the residuals, obtained from the optimal parameter sets, are evaluated over a set of 10 experimental replicates

Fig. 7.

Performance comparison of the fitting procedure applied to the TdDC model, with the steepness parameter n of the initial condition chosen among three values (, and ), and with n fixed (). For each fluid, the mean and standard deviation of the squared 2-norm of the residuals, obtained from the optimal parameter sets, are evaluated over a set of 10 experimental replicates

Fluid classification

The calibration procedure illustrated above allows us to identify each time series by a set of optimal parameters for the TdDC model, which reveal the transport properties of the observed electrical discharge. The parameters and D characterize two properties of transport, that pertain to the discharge seen as a flow through a narrow channel. In that case, two contributions combine to produce the total flow: the cross section of the channel, that determines the amount transported at a given flow rate, and the speed at which matter flows. In the TdDC model, the first is somehow related to D, the second to . The goodness of the results presented in this section indicates that our interpretation of the phenomena occurring in our experiments is correct: this decomposition of effects accurately describes our experiments. Reality may be more complex, for instance, one may expect that the discharge follows, like in lightnings, more than a single path, and that these paths are far from linear. However, this does not refute our view: there will be a maximum amount that can pass at fixed speed, and there will be a given speed, at least as average quantities. The complexity of the phenomenon, ultimately requires an anomalous transport description, with two parameters. The optimal parameter space resulting from the data fitting of the TdDC model highlights regions that can be used to classify fluids with different insulation properties.

Conductive and insulating fluids

We have collected experimental measurement of electrical discharge observed applying a 5 V electrical voltage to the micro-electrodes immersed in fluids with known insulating properties. Specifically, we used both insulating fluids, including air, Isopropyl alcohol, Sf33, Novec, Diala, Ethanol, and conductive fluids, namely tap water, Deionized water and Ultrapure water. For each fluid, we measured 10 replicates. The optimal parameter values, represented as coordinates in the parametric space , are shown in Fig. 8.

Fig. 8.

A Optimal parameters obtained for conductive and insulating fluids, obtained from fitting the data on the voltage discharge observed when a 5 V electrical voltage to the micro-electrodes is applied. B Zoomed-in views of the parameter space showing the optimal parameters obtained for insulating fluids

The repeated tests for each fluid are well localized in the parametric space, supporting that the fit results are robust. Notably, the optimal parameters exhibit a linear trend within the parametric space. The same behavior is observed at higher voltage configurations, such as 10 V, 12 V, 15 V, and 20 V. Specifically, at higher initial voltages, conductive fluids are associated with higher values of both the optimal D and parameters (see Fig. 9A). In contrast, insulating fluids do not exhibit significant changes in their position in the parametric space as the initial voltage varies (see Fig. 9B). Moreover, the color gradient in Fig. 9 shows that the successive trials do not align in an orderly manner along the trend line. This disorder indicates that the fluid does not deteriorate as tests are repeated, but it is more likely due to variations in the initial conditions, i.e. in the exact distribution of the charge in the electrodes at time .

Fig. 9.

The optimal parameters obtained for A Deionized water and B Novec from fitting the data on voltage discharge observed when an electrical voltage of 5 V, 10 V, 12 V, 15 V and 20 V is applied to the micro-electrodes. The successive tests are indicated by a color gradient from darker to lighter shades. This color code does not reveal any temporal ordering of the results. On the other hand, the exponential dependence of D on for the single fluids and for all of them is evident

The optimal parameters resulting from the discharge in insulating fluids are all located in a neighbourhood of , see also Table 1. This reflects a normal diffusion process with a low diffusion coefficient. Even if the values are quite similar it is worth noting that fluids with very good insulating properties, with dielectric strength higher than 30 KV in standard testing conditions, can be clearly distinguished from less insulating fluids like SF33. In particular the estimated value of the diffusion parameter of the SF33 is approximately two times the values of Novec or Diala insulating fluids, see Table 1. The result is very relevant from the point of view of the application because a non destructive technique, operated at low voltage, is predictive regarding the behaviour of the fluid at high voltage. The discharge in conductive fluids instead results in greater values of and D, or both, suggesting a super-diffusive process with a higher diffusion coefficient. The positioning of the optimal parameters in the parametric space thus suggests a robust method for classifying fluids with different insulation properties.

Table 1.

The average dielectric strength and its standard deviation are measured and compared with the mean values of the optimal parameters, and , evaluated across a set of ten experimental replicates

Fluid	Average dielectric strength (kV)	Std. dev.
Novec 7100	48.27	3.12	1.0020	9.1033e-05
DIALA S4 ZX-I	38.35	7.17	9.7849e-01	8.3962e-05
SF33	2.02	0.01	1.0337	2.0222e-04

Ethanol-water mixtures

The same procedure is applied when the micro-electrodes are immersed in different mixtures of ethanol and tap water, with ethanol concentration varying from to . The resulting optimal parameter sets are represented in the parametric space in Fig. 10A. The mixtures are clearly distinguished in the parametric space, except for those with less than ethanol in water, which are described by similar transport parameters and overlap in the parametric space, as required for fluids with same conductivity. Interestingly, the optimal parameters corresponding to mixtures with increasing percentages of ethanol in tap water do not follow the path of minimal distance in the parametric space. Conversely, they follow a specific functional form resembling a tilted parabola. The same functional trend is observed also for higher voltage initial configuration, i.e. 12 V and 24 V. In Fig. 10B, the centroids of the optimal parameters obtained from the ten repeated tests are fitted with a parabolic curve. Specifically, the fitting curve p(x) is determined by rotating the data by 10 degrees and applying a parabolic fit using MATLAB’s built-in polyfit function. The resulting polynomial is given by . This straightforward fit holds significant practical relevance, as it can be used where data are missing and, in particular, to predict the proportion of an unknown tap water–ethanol mixture, as discussed in the next conclusive section.

Fig. 10.

A Optimal parameters for mixtures of tap water and ethanol are obtained by fitting data on the voltage discharge observed when a 5 V electrical potential is applied to the micro-electrodes. B Centroids of the optimal parameters are rotated by 10 degrees (blue points) and fitted to a parabola (blue curve)

Numerical methods

Numerical simulations of the four PDE models are performed on the 1D spatial domain [0, 1], with initial and boundary conditions imposed at and , as specified in Eq. (2) (main text). The Gaussian Model with Time-Dependent Diffusion Coefficient, the Porous Medium Equation, and the Kardar-Parisi-Zhang Equation are solved using the MATLAB routine pdepe. The Telegrapher Equation is solved by discretizing the spatial derivative, and solving the associated system of two first-order ODEs (for u and its time derivative) using the MATLAB solver routine ode45. The performance ranking of the PDE models is determined by comparing the mean and standard deviation of the squared 2-norm of the residual R, defined in Eq. (7) of the main text. The residuals are evaluated using the MATLAB routine lsqcurvefit by fitting the models to ten replicate datasets for each fluid. This procedure results in the selection of the TdDC model with a fixed steepness n of the initial condition, specifically . All computations were performed using MATLAB R2020a.

Discharge data

Experimental setup and sample preparation

Because of the thin metallic layer of the micro-gap, it is impossible to directly wire it to the switch and the signal acquisition electronics (electrical switch, power supply and oscilloscope) without any risk of removing it. For this reason, during the preparation of the sample, a conductive paste is manually deposed on the square pads. Connections can then be ultimately sured by wiring or by contact spring connectors, see Fig. 11. For our tests we implemented a toggle switch On-Off-On, DPDT (Double Pole, Double Throw), supplier: RS PRO components.

Fig. 11.

A Connection between the micro-gap pads and spring connectors (also wired to the switch-capacitor-generator-oscilloscope circuit). The micro-gap and sample fluid are in a dedicated prototyping housing. B Detail image of the connection

Fluids specifications

Tables 2 and 3 show the values of dynamic and kinematic viscosity for each tested fluid.

Table 2.

Density, dynamic and kinematic viscosity values for tested insulating fluids

Unit	Shell DIALA	Novec	OPTEON	Ethanol	Isopropyl
	S4ZX-I	7100	SF33	96%	alcohol
Density (g/ml)	8.05	1.52	1.36	0.805 - 0.812	0.786
Dynamic	77.28 (+40C)	0.58 (+25C)	0.38 (+25C)	1.2 (+20C)	2.37 (+20C)
viscosity	3075.1 (−30C)
(mPa s)
Kinematic	9.6 (+40C)	0.38 (+25C)	–	1.51 (+20C)	2.61 (+20C)
viscosity	382 (−30C)
(mm/s)

Table 3.

Density, dynamic and kinematic viscosity values for tested conductive fluids (prepared mixtures with water are excluded)

Unit	Ultrapure water	Deionized water	Tap water
Density (g/ml)	0.9982	1.002	1
Dynamic	1.0016 (+20C)	1.002 (+20C)	1.0016
viscosity
(mPa s)
Kinematic	1.002 (+20C)	1.0038 (+20C)	1
viscosity
(mm/s)

Nominal Dielectric Strenght values of tested insulating fluids (kV):

• SHELL DIALA S4 ZX-I: ;

• 3 M NOVEC 7100: ;

• OPTEON SF33: 10.

Water conductivity (S/cm, room temperature):

• Ultrapure water: 1.92±0.001;

• Deionized water: 32.95±0.001;

• Tap water: 415.31±0.001.

Conductivity values were taken using a watertight portable conductivity meter (CA 10141 model, produced by Chauvin Arnoux).

Data reproducibility

All measurements were conducted under room temperature and humidity conditions, with repeatability assessed through ten consecutive measurements for each sample fluid.

Figure 12 shows the mean value and standard deviation of the measured discharge, evaluated across ten replicates for each fluid. As a representative case, results are shown for an initial voltage of 12 V applied to the micro-electrodes. The discharge process is highly reproducible. Similar trends are observed for initial voltages of 5 V, 10 V, 15 V, and 24 V. The reported degree of reproducibility supports the predictive ability of the approach described in this paper which, in turn, indicates that the essential physics of the phenomena investigated here has been properly described.

Fig. 12.

Mean and standard deviation of the measured discharge, evaluated across ten replicates (each consisting of ten thousand time points) for each fluid, with an electrical voltage of 12 V applied at the micro-electrodes

Discussion

Continuous and real-time monitoring of fluid properties is essential for industrial applications, as it enables predictive maintenance of equipments with significant improvement in fluid waste reduction and equipment safety. The class of sensors developed by Eltek used in this study surpasses traditional techniques by employing micro-gaps as electrodes, allowing it to operate at low voltages (5-12 V), also on small liquid volumes (few microliters are generally sufficient), in a compact miniaturised package that can be adapted to several environments, see Fig. 13. We adopted an experimentally-informed mathematical modeling approach to investigate the anomalous behavior of the electrical discharge observed within the mentioned sensor immersed in fluids with varying insulating properties. From a modeling perspective, we focused on four models consisting of partial differential equations to describe the discharges: the Gaussian Model with Time-Dependent Diffusion Coefficient, the Porous Medium Equation, the Kardar-Parisi-Zhang Equation, and the Telegrapher Equation. Data fitting revealed that the Gaussian Model with Time-Dependent Diffusion Coefficient most accurately replicates the experimental dataset. Rarely, one of the other models approaches or barely surpasses the performance of the TdDC. However, that is not enough to make any of them the best choice. Also, the use of a variety of models, instead of a single one, would make problematic the applicability of the method to the development of sensors, while offering no real performance improvement. Our choice makes the optimal parameters not only elucidate the anomalous transport properties of the electrical discharge in the specific fluid but also suggest a novel method for fluid classification that is robust to both replicate measurements and variations in the applied initial voltage. Additionally, we applied the proposed approach to ethanol-water mixtures, demonstrating the potential for accurate predictions on the composition of unknown relative percentages in fluid mixtures. This predictive ability indicates that the present approach encompasses the essential physical ingredients of the phenomena that have been investigated, in terms of the diffusion coefficient D and of the transport exponent .

Fig. 13.

A Micro-gap sensor: assembled. B Opened, with electronics

The industrial application of the approach is under evaluation not only for liquid mixtures classification but also for gas and gas-liquid mixtures. For example, the simplest application of the device in gas-liquid mixtures is the rapid measurement of water-air mixtures, acting as a fast humidity sensor. The described mathematical approach is indeed as simple as powerful and is ready to be applied when a moderate computing power is available. Indeed, the simple positioning of the model’s optimal parameters in the parametric space can uncover differences in the insulative or conductive properties of fluids that conventional techniques may fail to detect. Furthermore, as seen with the specific tap water-ethanol mixture, this approach can be extended for predictive purposes: the fitting results can help in providing insights into the fluid’s composition. The ongoing effort is aimed at the further simplification of the required electronic hardware by the implementation of a low cost miniaturised electronics based on neural networks. The training of the networks, and all the related needed power of calculation, will be done in advance on a wide experimental and synthetic dataset, generated by the presented model. Once trained, the networks will be implemented on low cost electronics, enabling the production of sensors operating autonomously and not necessarily connected to remote processing units.

Author contributions

S.B.: conceptualization, formal analysis, investigation, methodology, data curation, software, validation, visualization, writing–original draft, writing–review and editing. P.B.: conceptualization, data curation, investigation, methodology, visualization, writing–review and editing; M.P.: conceptualization, investigation, data curation, methodology, supervision, writing–review and editing; L.R.: conceptualization, formal analysis, funding acquisition, investigation, methodology, supervision, writing–review and editing.

Funding

This work is part of the project NODES which has received funding from the MUR—M4C2 1.5 of PNRR funded by the European Union—NextGenerationEU (Grant agreement no. ECS00000036).

Declarations

Competing interests

We declare we have no conflict of interest.