A mathematical model of OECTs with variable internal geometry

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Highlights

  • As ions of a system are dislodged to the electrodes, the internal configuration of an OECT can vary.

  • A mathematical model predicts the fraction of the system that is permeated with the ions.

  • The same model predicts the concentration of the ions in the system.

  • The model receives as an input the values of current and voltage generated by the OECT.

  • The model enables quantification of a system without calibration of the device.

Abstract

Organic Electrochemical Transistor (OECTs) are devices that can measure the ionic content of liquid samples and biological systems. The response of an OECT can provide information on the physiological conditions and characteristics of a biological system. In a typical OECT configuration, the system or sample is connected to a reference electrode (the gate) and to a semiconducting material, typically PEDOT:PSS, with two other terminals (the drain and the source) for connection to an external circuit. The transistor architecture of OECTs enables signal control and amplification. Upon application of an external electromagnetic field at the electrodes, ions are driven from the liquid sample towards the PEDOT:PSS channel, where they modify the conductivity of the channel and generate a continuous current as a function of time. The intensity of that current and the time to the steady state can be correlated to the characteristics of the ions in solution. In most of the existing theories that model the behavior of OECTs, the internal configuration and geometrical parameters of the device are assumed to be constant over time. This simplifying assumption breaks down in living systems and in all those soft devices with elevated value of compliance and absorption (such as devices on paper, textile or polymeric sponges). Similar simplified models may fail to predict the behavior of real systems within acceptable bounds. Here, we present a mathematical model that describes the behavior of OECTs in which the geometry of the internal fluidic circuits of the system can change over time. These circuits represent the network of chambers and channels through which the liquid solution flows from the gate to the drain-source electrodes, enabling the transport of ions. At a certain time, the liquid solution shall be spread throughout a fraction only of the entire network available for liquid transport, i.e. the wet fraction p. The mathematical model that we have developed in this work uses the data generated by OECTs to determine the wet fraction p and the concentration C of ions of a system. The model enables quantification of a system without calibration of the device, which may be of interest for those working in the fields of bioengineering, biomedical sensors, wearable electronics, flexible electronics. In experiments where the variables of system were varied over large intervals, the model achieved an excellent performance and a precision up to 92%.

Introduction

Organic electrochemical transistors (OECTs) are devices based on a semiconductor, typically an organic polymer, that is permeable to the ions of a solution and can be doped/dedoped by those ions under the action of an external voltage. The change of the bulk conductivity of the entire device is indicative of the physical and chemical characteristics of the solution. State-of-the-art OECTs are mostly based on the conducting polymer poly(3,4-ethylenedioxythiophene) doped with polystyrene sulfonate (PEDOT:PSS) [1].

The signal measured by an OECT is a current I that smoothly transitions from zero (no flux) to a steady state value of current, similar in shape to exponential function of the type Im1-e-t/τ, where t is time, and m and τ are the modulation and time constant of the system. While the current I measured by an OECT is proportional to the concentration and characteristics of ions in solution, mathematical models provide ways to decipher the correlation between I and the ions characteristics, including concentration, charge, and size [[2], [3], [4]]. The combination of an OECT with convenient methods of data analysis represents a platform for advanced sensing applications of complex and biological systems. This platform can convert the biophysical signature of a system into an electric signal, with clear advantages over conventional chemical, spectroscopic or optical methods of analysis. Electric signals obtained from the OECT can be easily amplified, processed, and brought to other devices or computers for recording and analysis. Moreover, the OECT can be controlled with a very small level of power and, with the advent of IC (integrated circuit) technology and nanotechnologies, the device is susceptible of miniaturization, with reduced costs, reduced materials consumption, improved precision and selectivity, enhanced portability [5]. Moreover, OECTs exhibit the attributes of biocompatibility, facile deposition, high sensitivity, high signal to noise ratio [[6], [7], [8]].

Biosensors based on an OECT architecture have been used – to cite a few – for electrophysiological recording, bio-sensing applications and applications at the bio-interface [1,9,10], bio-computing [11], neuromorphic engineering [[12], [13], [14]], as biosensors to monitor in real time the physiological characteristics of tomato plants [15], as a sensor for cells [16]. Other technological solutions involve the integration of OECTs with functional substrates and textile fibers, and have been used to monitor biological fluids in wearable solutions, such as the monitoring of human sweat [17,18].

In all cited reports and in the existing deterministic models of OECT operation, the characteristics of the system are determined under the simplifying assumption of fixed geometry of the device. That is, it is assumed that the ions originally dispersed in the electrolyte are transported to the active electrode of the device through an ideal pathway with cross sectional area and conductivity that do not vary with time. This automatically implies that the composition, physical and chemical characteristics, of the portions of the device through which the ions in the solution and the electrodes communicate, remain constant over time. This hypothesis may be appropriate for closed systems, i.e. devices that incorporate all necessary steps for sample analysis (sampling, sample transport, filtration, dilution, chemical reactions, separation and detection) like micro total analysis systems (micro TAS), microfluidic chips, lab-on-a-chips, performing an ex vivo analysis of samples. However, the fixed geometry assumption breaks downs for all those devices that are integrated in the biophysical system that they measure, performing an in vivo analysis of samples. We shall give some examples to illustrate the case.

In OECTs designed to measure the physiological conditions of plants, parts of the plant turn to be active constituents of the devices themselves. In those systems, water, dissolved minerals and ions are conveyed from the roots of the plant to the electrodes of the OECT device through the plant vascular tissue, or xylem. The xylem is the plant vascular tissue that conveys water and dissolved minerals from the roots to the rest of the plant, providing physical support. Xylem tissue consists of a variety of specialized, water-conducting cells. Being a living material, the conformation, resistance and conductivity of the xylem may change over time, depending on the sap content of the plant, and in turn may depend on external factors, such as irrigation [15]. (The sap is fluid transported in the xylem, it is the whole of water, inorganic and organic nutrients transported through the plant.) The sap content may be different in different vessel elements (trachea) of the plant vasculature. Thus, the transport of species in the system is time and space dependent; variations in the response of the OECT can reflect this dependence. The scheme reported in Fig. 1a describes a similar system. Ions are transported from a reference electrode (the gate) to the PEDOT:PSS channel, contacted to the source and drain electrodes. In an ideal representation of the system, ions travel through independent vessel elements, each of which has specific values of sap amount, thus each of those vessels has different values of resistance. The overall resistance depends on the number of vessels that are wet from sap to the total number of vessels in the vasculature (i.e. the wet fraction p), and on the concentration of ions in those vessels (C).

Wearable sensing devices represent another example of a system where the internal values of resistance are not constant. Consider for sake of clarity the scheme in Fig. 1b. Here, the device is a patch put in direct contact with the skin of a patient. The patch is a fabric of threads where are inserted, from the outside, the gate, and a thread functionalized with the conducting polymer PEDOT:PSS that is in turn contacted to the drain and the source. Because of the externally applied voltages Vgs and Vds, ions in the patch are transported from the system to the PEDOT thread, and through that to the source, where they generate a current, Ids, i.e. the response of the system. The total charge that arrives at the source in the unit time would depend on the intensity of the externally applied voltages (that is a controllable parameter of the system) and on the resistance that the patch offers to the flow of ions. Since the resistance depends on the physical and chemical characteristics of the materials that in turn may depend on its hydration status, it may not be constant in space or time. The more the material is saturated with water or another liquid, the more the resistance may deviate from its normal values measured in dry conditions. In other terms, as the patch gets wet from sweating - under normal operating conditions of the devices - the values of electric conductivity of the system may vary, either because its internal elements are permeated with water or sweat, i.e. the solvent, or because that solvent may more easily solubilize additional ions or charged species in its volume. This variation may be space and time dependent: the amount of resistance change shall depend on the ducts in the skin that release sweat (where) and on rate at which they do it (when). Assuming that the resistance is constant represents an oversimplification that breaks down in real biological systems and may lead to incorrect results.

The transport and detection of charged species in a system is a mechanism described by equations that involve coupled variables, among others: the water content of the system (i.e. the wet fraction, p) and the concentration of ions in that system (i.e. C), being both a function of space and time.

Here, we have developed a mathematical model that describes the behavior of OECTs as a function of C and p. The model correlates microscopic (C) and macroscopic (p) variables to the physical observables of the system, i.e. applied voltage and current. Then, using a numerical scheme, we decouple variables and provide a solution for p and C. Results were validated by experiments. The model may be used to examine biological systems even without direct knowledge of its internal workings.

Section snippets

The physical model

Consider an OECT device integrated in a biophysical system (Fig. 2a). The electroactive species of interest are dispersed in an electrolyte initially contained in the biophysical system, and contacted to the device through a reference electrode (the gate) and a conductive polymer channel, typically PEDOT:PSS, that in turn connects the electrolyte to the source and drain electrodes. Upon application of a voltage between the drain and the source (Vds), and the gate and the source (Vgs), currents

Numerical solution of the model

Eq. (2) involve functions of the variables combined in a non-algebraic way, so that direct inversion of the system and closed form solution of the variables are not possible. One can find a solution for the unknown variables in Eq. (2) using a numerical scheme. To do this, firstly we recast Eq. (2) as:ψi=ξi,i=1,,3whereψ3=Igs,andξ1=Vds/Rdsdry+Ro/modC+1-Rdsdry/Nn,ξ2=Vds/Rdsdry+Ro-Rdsdry/Nn,ξ3=Vgsϕ/ρCln.

Thus, the terms ψi are associated to values of current measured by the OECT, while the

Validation of the model

To assess the mathematical model’s capability to be predictive in nature, we benchmarked the model against experimental data acquired using an OECT device with a controlled design and known physical characteristics. The device was used to measure NaCl ions in solution. Details on how the device is made and its implementation are reported in the Methods. A picture of the real device is reported in a separate Supporting Information #5. In the experimental set-up, we fix he fraction of the

Discussion

High values of standard deviation registered for pe=1 indicate a reduced reproducibility of the output of the model. That in turn indicate that the measured values of Idsmin, Idsmax and Igs, from which the concentration and the wet fraction are derived, oscillate more energetically around the mean. Uncertainty in the response of the system at elevated values of wet fraction approaching unity may be indicative of the fact that the PEDOT channel absorbs less efficiently the liquid solution. The

Conclusions

The mathematical model that we have developed in this work enables to determine the wet fraction p and the concentration of ions C of biological systems from the read-out of an OECT device. The model receives as an input the current and voltage values measured by the device, and yields as output the values of concentration C and wet fraction p of the system, upon minimization of a cost function Y. In test experiments with artificial analogues of a biological system, we have determined the

Fabrication and operation of the OECT for the validation of the model

To validate the model, we compared the predictions of the model with the output of an OECT system implemented in a system with known physical and chemical characteristics. The OECT device is based on the PEDOT:PSS conducting polymer thin film channel, deposited on a textile fiber, placed in direct contact with an electrolyte and with a gate electrode immersed in it. A source–drain voltage (Vds) is applied at channel terminals, generates a drain current (Ids), which drives holes along the

CRediT authorship contribution statement

Francesco Gentile: Formal analysis, Investigation, Methodology, Supervision, Validation, Visualization, Writing - original draft. Filippo Vurro: Data curation, Investigation, Methodology. Francesco Picelli: Data curation, Investigation, Methodology. Manuele Bettelli: Data curation, Investigation, Methodology, Writing - review & editing. Andrea Zappettini: Funding acquisition, Resources, Supervision, Writing - review & editing. Nicola Coppedè: Conceptualization, Data curation, Funding

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Francesco Gentile is an Associate Professor in the Department of Electric Engineering and Information Technology, University Federico II of Naples, Italy. Prof. Gentile does research in biomedical nanotechnology. He uses mathematical modelling and nanotechnologies to engineer solutions to biomedical problems. Prof. Gentile authored more than 100 papers in peer-reviewed journals, and he is the inventor of 5 issued patents. Prof. Gentile earned his MS in Mechanical Engineering at the University

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Francesco Gentile is an Associate Professor in the Department of Electric Engineering and Information Technology, University Federico II of Naples, Italy. Prof. Gentile does research in biomedical nanotechnology. He uses mathematical modelling and nanotechnologies to engineer solutions to biomedical problems. Prof. Gentile authored more than 100 papers in peer-reviewed journals, and he is the inventor of 5 issued patents. Prof. Gentile earned his MS in Mechanical Engineering at the University of Calabria in 2003, and a PhD in Biomedical Engineering at the University Magna Graecia of Catanzaro, Italy, in 2008.

Filippo Vurro is a third year PhD student in Material Science and Technology at IMEM-CNR. His doctoral research is focused on the development of electrochemical biosensors for plant phenotyping and smart agriculture. He got a Bachelor's degree in Biology and a Master’s degree in Bio-molecular Chemistry from the University of Parma.

Francesco Picelli has a bachelor and master degree in Chemistry obtained at the University of Parma, for theses has worked on the characterization of OECT, based on PEDOT:PSS soaked on textile thread, at National Research Council of Italy at IMEM facility. Now he's a PhD student in Materials Science and Technology at the Institute of science and technology for ceramics, facility of National Research Council of Italy, working on transparent ceramics for LASER application.

Manuele Bettelli carried out his Master Degree in “Physics” (Physics department - Università degli Studi di Parma) and he graduated in July 2014. He worked for three years at IMEM-CNR and he defended the Ph.D. thesis in March 2018 (Material Science and Technology). He actually works as researcher in SIGNAL groups, IMEM-CNR (PR). Since 2014, he co-authored 22 papers in international journals and sent contributions to 17 international conferences. H-index is 7 with 101 total citations (Google Scholar).

Andrea Zappettini is a senior researcher at IMEM-CNR. He mainly developed sensors based on novel and multifunctional materials. He is author of more than 200 scientific papers on international journals, cited more than 2500 times, h-index 28. He is also author of 12 International patents and 5 National patents.

Nicola Coppedè Nicola Coppedè graduated in Physics at the University of Pisa in 2001 and obtained the PhD degree in Physics at the University of Trento in 2006. He was selected for a permanent position as researcher in IMEM CNR Parma in 2012. His research activity focuses on organic biosensors, in particular on functional substrates, like textiles, carbon fibers, polymeric sponges, for biomedical and physiological applications. He developed innovative devices, in particular textile biosensors dedicated to the monitoring of human physiological fluids, biocompatible fiber sensors for plant sap monitoring and polymeric pressure sensors. He is expert of the material functionalization with organic, metal oxide and nanohybrid materials. He published more than 85 papers in international journals and invented 6 issued patents.

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