A Mechanical–Electrical Model to Describe the Negative Differential Resistance in Membranotronic Devices

Membranotronic devices are artificial neural membranes mimicing the functionality of biological neural networks. These devices rely on the emergence of negative differential resistance (NDR). A minimalistic physical model for membranotronic devices capable of generating NDR is presented. The model features a deformable membrane with holes that facilitate ion currents. The deformation of the membrane, induced by electrostatic pressure from an applied voltage, modulates these currents. The model comprises a well‐established mechanical framework for describing deformable membranes with holes, alongside a model for ionic current that considers temperature‐dependent ion mobilities. It is demonstrated that the model can faithfully reproduce NDR across a wide and physically realistic range of parameter combinations. Furthermore, the simulations reveal that the temperature of the electrolyte can exceed its boiling point, resulting in bubble formation. To mitigate this issue, materials with high heat transfer coefficients and low conductivity are recommended. In essence, the work bridges the gap between artificial membranotronic devices and biological neural networks by providing a robust physical model capable of emulating NDR, a key feature in the operation of such systems. This advancement in membranotronics holds great promise for the development of bioinspired soft artificial neuromimetic systems that closely mimic their biological counterparts.


Introduction
In the realm of artificial systems, the fixed and discrete nature of functional blocks has traditionally limited self-reconfigurability and the potential intelligence of such creations.In contrast, the natural world offers a profound lesson in adaptability, where functionalities are delocalized, and shapes continuously evolve to suit the environment, optimizing energy usage and spatial efficiency.
One example of this adaptability is found in universal neuronal systems, where cellular membrane shape, synapse location, and synapse type directly dictate functionality.It is in this spirit that the concept of membranotronics emerges, [1] envisioning soft systems reliant on shape reconfigurability, thus influencing signal propagation and information processing functions.Much like the dynamic variations in natural systems, including compression/expansion, wrinkling, and curling [2] enable adaptation to changing environmental conditions, membranotronics aims to emulate this behavior.
The distinctive electromechanical behavior exhibited by ion transport channels in a biological neural membrane underlies its unique functionality, playing a crucial role in signal generation, transmission, and the overall operation of neuronal cells.[5][6] As a result, there is a burgeoning interest in the creation of analog signal processing units, leveraging a combination of electronic [7] and ionic gating and transport principles. [8]Within these principles, the exploration of ionic transport through nanopores has gained prominence due to significant parallels with natural ionic channels, closely Membranotronic devices are artificial neural membranes mimicing the functionality of biological neural networks.These devices rely on the emergence of negative differential resistance (NDR).A minimalistic physical model for membranotronic devices capable of generating NDR is presented.The model features a deformable membrane with holes that facilitate ion currents.The deformation of the membrane, induced by electrostatic pressure from an applied voltage, modulates these currents.The model comprises a well-established mechanical framework for describing deformable membranes with holes, alongside a model for ionic current that considers temperature-dependent ion mobilities.It is demonstrated that the model can faithfully reproduce NDR across a wide and physically realistic range of parameter combinations.Furthermore, the simulations reveal that the temperature of the electrolyte can exceed its boiling point, resulting in bubble formation.To mitigate this issue, materials with high heat transfer coefficients and low conductivity are recommended.In essence, the work bridges the gap between artificial membranotronic devices and biological neural networks by providing a robust physical model capable of emulating NDR, a key feature in the operation of such systems.This advancement in membranotronics holds great promise for the development of bioinspired soft artificial neuromimetic systems that closely mimic their biological counterparts.
[16] The design of 3D pores, including conical shapes inspired by biological ion channels in neuronal cells, has been a focal point.19] Modified poly-dimethyl siloxane (PDMS) has been used as the material of choice, boasting exceptional mechanical and electrical properties. [20]This class of electromechanically active materials with integrated nanoscale holes is synthesized and scrutinized for their ability to enable excitation, inhibition, and signal propagation along the surface which is electrostatically polarized.This leads to an electrostatic pressure that deforms the membrane and, hence, the integrated holes close or open depending on the state of the polarization, as shown in Figure 1.The membrane, serving as an elastic nonlinear electrochemical waveguide, integrates a multitude of electromechanically gated ionic channels.As the signal propagates, mechanical pressure and electrostatic polarization across the membrane lead to localized regulation of ionic transport.The core of signal propagation in membranotronics is the electrochemical pulse, which maintains its shape over significant distances, akin to the concept of an action potential promoted by negative differential resistance (NDR), which plays a crucial role in the functioning of neural membranes. [21]ere, we develop a simple physical model which reproduces the NDR reported for membranotronic devices.For this, we employ the mechanical model for a micro-electro-mechanical system (MEMS) to simulate a deformable membrane with a cylindrical hole.Through this hole ionic current can flow and the strength of this flow in dependence of applied voltage is calculated.The temperature effects on the mobility of the ions are also considered.
We structured our article as follows.In the next section, we set up the model by introducing its different parts for the deformable membrane, the ionic current, and the temperature dependence of the ion mobilities, respectively.Then, the results of a parameter variation study are used to compare the simulation with experiments found in the literature.Finally, we summarize our work and give an outlook on how to improve the model further.In this context, membranotronics represents a groundbreaking endeavor to replicate the remarkable behavior of neural membranes in artificial systems, offering exciting possibilities for the future of adaptable, intelligent, and bioinspired technology.

Model
In the following, we will delve into the various components comprising our physical model, each of which plays a pivotal role in faithfully replicating the current-voltage characteristics observed in membranotronic devices.Our exploration commences with an examination of the membrane itself, subjected to deformation under the influence of electrostatic pressure.This deformation induces voltage-dependent alterations in the geometry of the holes within the membrane structure, consequently exerting control over the flow of ionic current through these apertures.Subsequently, we turn our attention to the crucial factor of temperature dependence pertaining to ion mobilities, recognizing its significance in shaping the behavior of our model.

Deformable Membrane
24] The system is sketched in Figure 2. Balancing of the electrostatic force acting on the capacitor plates and the restoring force originating from the spring leads to a relation between the deflection x of the movable plate and the applied voltage where d 0 is the initial distance of the plates (for V ¼ 0), ε the permittivity of the medium between the plates, A p the surface area of the plates, and k the spring constant.The solution of this force balance is In the following, we will identify the medium between the plates with the membrane.Hence, its thickness is given as Using the definition of Young's modulus Y as quotient of tensile stress and axial strain, the spring constant can be calculated as k ¼ YA p =d 0 .If we assume that the membrane has a cylindrical shape with initial radius R 0 and that the hole is also cylindrical with initial radius r 0 and located in the center of the membrane, the hole radius is given as where ν is Poisson's ratio of the membrane.Note that in the derivation of Equation ( 4), we approximated the small segment of the cylindrical disk as slab of thickness d 0 and length l 0 and applied the definition of ν. [25] Based on Equation ( 2), (3), and (4), the geometry of the hole is defined for a given applied voltage.

Ionic Current Through the Hole
The ionic current through the hole is given as where e is the elementary charge and A h the hole surface area.
For each ion species i, the charge number Z i , mobility μ i , and number density n i are needed.
In the remainder of this article, we will use an aqueous KCl solution as electrolyte.Therefore, The sum of the mobilities of the K þ and Cl À ions can be calculated as where κ is the electrolytic conductivity, and Equation ( 5) simplifies to At this point, we want to state that Equation ( 5) is only valid for hole surfaces A h that are large enough to neglect edge effects on the electric field as it is calculated based on the approximation of a parallel plate capacitor.More specifically, the electric field between the two infinitely extended plates was used during the derivation.Therefore, effects caused by the finite and small dimensions of the geometry like edge effects or the electrical double layer are omitted.Obviously, these effects will play a role because the hole is closing.To overcome this, the field can be calculated numerically based on Poisson's equation.However, this approach is very elaborate and part of a future work because Equation ( 5) is also suitable to reproduce the NDR which will be shown in later parts of this work.

Temperature Dependence of the Mobilities
To calculate the temperature T of the electrolyte, we consider Joule heating due to the current flow and heat transfer from the electrolyte to the membrane where h is the heat transfer coefficient, A m the shell surface of the hole, and T m the temperature of the membrane which we assume to be constant.Balancing these two equations leads to where Equation ( 7) was used as expression for I.

Results and Discussion
In this section, we will introduce the default parameters for the model described earlier.Afterward, the performed parameter variations are presented and discussed.Finally, we will compare the simulation results with experimental data found in the literature.

Default Parameters
In this article, we study a membrane made of PDMS.The electrolyte is an aqueous solution of KCl.If not stated otherwise, we use the parameters given in Table 1.
We did not find any data relating to the heat transfer coefficient of aqueous KCl solution and PDMS.Measurements by Maia et al. [26] show h ≈ ð100 : : : 1300ÞW K À1 m À2 for distilled water in PDMS microchannels, so we decided to take h ¼ 1300 W K À1 m À2 as default value and perform a variation of this parameter to study the influence on the current.
In general, the electrolytic conductivity κ depends on temperature.We fitted the data for M ¼ 1 mol kg À1 given by Haynes [27] to a cubic function; the result is shown in Figure 3.For this case, the analytical solution of Equation ( 10) is given in Appendix.

Variation of Initial Hole Radius
First, we want to study the model for a constant temperature of 25 °C.Figure 4 shows the current as well as the relative hole surface area in dependence of applied voltage for different initial hole radii.As expected, holes with increasing r 0 need higher voltages to be closed again and, hence, higher currents are possible.The essential functionality needed for membranotronic devices, the NDR, is already present in this very basic model with constant electrolyte temperature.
At the moment, the hole itself is assumed to be of cylindrical shape.The deformation of arbitrary hole geometries can be simulated using finite element models and the results can then easily be incorporated into the existing model, which will be part of a future work.In this way, it is also possible to include buckling of the membrane which can, depending on the shape of the hole, lead to an increase of the effective diameter for increasing pressure.

Influence of Variable Temperature
Figure 5 shows the current-voltage characteristic and the temperature of the electrolyte for the model including Equation (10).The dashed lines display the data for T ¼ 25 °C ¼ const: (cf., Figure 3).As the radius of the hole is independent of T, the hole closes at the same voltage.The temperature influence is negligible for r 0 ≤ 5 nm because T varies only up to about 6 K.As r 0 increases, the IV curves become asymmetric and differ more from the curves with T ¼ const: For r 0 ≥ 50 nm, temperatures above 300 °C are reached.Note that our model does not include the effect of exceeding the boiling temperature of the electrolyte.Therefore, the formation of bubbles is not considered and one has to be careful to draw conclusions from simulation results with T > 100 °C. Figure 3. Data given by Haynes [27] and cubic fit of electrolytic conductivity κ of aqueous KCl solution in dependence of temperature T for M = 1 mol kg À1 .We also tried to include the effect of the heating of the electrolyte itself according to where C is the specific heat capacity and m ¼ ρA h d the mass of the electrolyte (inside the hole) with density ρ.Following Toner et al. [28] C does not depend on T. The density was fitted to a cubic function using data provided by Hnedkovsky et al. [29] Balancing Equation ( 8), (9), and ( 11) leads to This equation has to be solved numerically.All temperaturedepend results shown in this article were calculated with both temperature models, i.e., Equation (10) and (12).The temperature profiles did not show any difference (not shown here); hence, Equation ( 11) is negligible and Equation ( 10) models the temperature of the electrolyte sufficiently.

Variation of Heat Transfer Coefficient
The influence of the heat transfer coefficient h on the IV curves as well as the temperature for r 0 ¼ 10 nm is shown in Figure 6.As noted above, experimental values for h are only available for distilled water in PDMS microchannels but not for aqueous KCl solutions.Decreasing h leads to increasing T (because the membrane becomes a worse heat sink) as well as a shift of the maximum of the IV curve to lower voltages.For h < 900 W K À1 m À2 , temperatures above 100 °C are reached.

Variation of Electrolytic Conductivity
Figure 7 shows the influence of the electrolytic conductivity κ on the IV curve as well as the temperature of the electrolyte.Increasing κ by a factor of 10 (dashed lines) leads to higher currents.The shape of the IV curves becomes asymmetric for smaller values of r 0 and the maximum of the current is shifted to smaller voltages.The temperature in the electrolyte Figure 5. Variation of the initial hole radius r 0 : current-voltage characteristic (top) and temperature of the electrolyte (bottom).The dashed lines correspond to the model with T ¼ 25 °C ¼ const: (cf., Figure 4).As the maximum temperature for r 0 ¼ 1 nm is only 25.2 °C, the curve can hardly be seen in the bottom plot.Figure 7. Variation of the initial hole radius r 0 : current-voltage characteristic (top) and temperature of the electrolyte (bottom).The solid lines mark the results with the original conductivity (cf., Figure 4).The dashed lines correspond to a conductivity increased by a factor of 10 and the dotted lines correspond to a conductivity decreased by a factor of 0.1.increases and the shape of the temperature curve becomes more rectangular for higher values of r 0 .Analogously, decreasing κ by a factor of 0.1 (dotted lines) leads to a decrease in current and temperature.Note that for simulations with T ¼ const:, increasing or decreasing κ only shifts the current along the I axis but does not influence the shape of the IV curve (not shown here).

Comparison with Experimental Data
Most of the default model parameters shown in Table 1 were chosen to match the experimental situation described by Faghih et al. [1] In contrast to our simulation, the membranes in the experiments usually have eight holes through which current is flowing, but there are also single-hole devices.For a wide range of parameters, the NDR is clearly reproducible with our model.Depending on the initial hole radius r 0 , maximum currents of up to 0.44 μA are calculated, as shown in Figure 4.However, the measured currents are up to two orders of magnitude higher than the simulated ones, which cannot solely be explained with the higher number of holes in the experiment.
Faghih et al. [1] reported the use of conical holes with diameters up to 600 μm and the measurements which show the NDR were done in the voltage range from À20 to 20 V.In our simulations, a closure of the hole in this voltage range is only possible for r 0 ≤ 20 nm (not shown here), which we attribute to the formation of a nanocrack rather than a hole as it was designed.
As shown in Figure 5, the temperature of the electrolyte increases with increasing initial hole radius and exceeds 100 °C for r 0 > 12 nm (not shown here).The effect of increasing temperature is even higher for lower heat transfer coefficients (see Figure 6).As the electrolyte is an aqueous KCl solution, T ≥ 100 °C leads to boiling and bubble formation.Currently, these effects are not considered in the simulation.However, Faghih et al. [1] reported that boiling of the electrolyte did occur during the measurements.T could be reduced by choosing materials with high heat transfer coefficients or electrolytes with lower conductivities (cf., Figure.6 and 7), respectively.It could also be worthwhile to explore liquid ionic solutions which have substantially higher stable operating temperatures. [30]e have shown that our model is capable to reproduce the essential functionality of membranotronic devices, namely, the NDR.Therefore, it is suitable for optimizing parameters in future experimental studies and devices.

Summary and Outlook
We present a straightforward yet effective physical model designed to replicate the observed NDR behavior in membranotronic devices.Our model centers around a deformable membrane featuring a central aperture, allowing the passage of an ionic current.To account for the temperature-dependent ion mobilities, we integrate various heat sources and sinks into our framework.
Through an extensive parameter variation analysis, we discern a logical trend: as the initial radius of the aperture increases, the voltage required for closure also rises, consequently augmenting the current.Furthermore, the introduction of temperature effects introduces an interesting nuance, resulting in asymmetric current-voltage curves, with the maximum current shifting toward lower applied voltages.Notably, our simulations reveal the potential for electrolyte temperatures to exceed their boiling points, particularly under conditions of high initial radii, limited heat transfer coefficients, and elevated electrolyte conductivities.
We benchmarked our simulation outcomes against experimental data obtained by Faghih et al. [1] revealing a commendable qualitative alignment.However, disparities emerge regarding the maximum current values and the precise dimensions and shapes of the apertures.It is imperative to acknowledge that our current model operates within a 1D framework, founded on elementary physical approximations.Notably, we have focused primarily on cylindrical apertures and adopted a simplified model for the ionic current within these apertures.Despite these simplifications, there are avenues for enhancing the model's fidelity, particularly concerning ion number density.Currently, we assume a constant ion concentration throughout the system.However, Equation ( 5) clearly illustrates that current varies with ion concentration.A 2D simulation incorporating concentration dynamics coupled with fluid flow could provide insight into the concentration profiles within the aperture.
In forthcoming research, our aim is to extend the model's capabilities by simulating multiple interacting apertures within a membrane.This expansion will enable the study of pulse generation and propagation, bringing us closer to emulating the intricate behavior exhibited by biological neural membranes.
the temperature is given as

Figure 1 .
Figure 1.Membrane with ionic channels which open and close depending on the applied voltage and, hence, the polarization of the membrane.a) Depolarized (relaxed) state of the membrane; b) Polarized (compressed) state of the membrane.

Figure 2 .
Figure 2. Sketch of the simulated geometry: parallel plate capacitor with a movable plate attached to a spring (top) and cylindrical membrane with a hole (bottom).The symbols are explained in the text.

Figure 4 .
Figure 4. Variation of the initial hole radius r 0 for T ¼ 25 °C ¼ const:: current-voltage characteristic (top) and quotient of hole surface area and initial area (bottom).

Figure 6 .
Figure 6.Variation of the heat transfer coefficient h for r 0 ¼ 10 nm: current-voltage characteristic (top) and temperature of the electrolyte (bottom).

Table 1 .
Standard parameters used to solve the model.