A mathematical model of OECTs with variable internal geometry
Graphical abstract
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 that smoothly transitions from zero (no flux) to a steady state value of current, similar in shape to exponential function of the type , where is time, and and are the modulation and time constant of the system. While the current measured by an OECT is proportional to the concentration and characteristics of ions in solution, mathematical models provide ways to decipher the correlation between 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 ), and on the concentration of ions in those vessels ().
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 and , ions in the patch are transported from the system to the PEDOT thread, and through that to the source, where they generate a current, , 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, ) and the concentration of ions in that system (i.e. ), 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 and . The model correlates microscopic () and macroscopic () 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 and . 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 (), and the gate and the source (), 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:whereand
Thus, the terms 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 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 indicate a reduced reproducibility of the output of the model. That in turn indicate that the measured values of , and , 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 and the concentration of ions 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 and wet fraction of the system, upon minimization of a cost function . 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 () is applied at channel terminals, generates a drain current (), 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
References (20)
- et al.
A theoretical model for the time varying current in organic electrochemical transistors in a dynamic regime
Org. Electron.
(2016) - et al.
A hybrid living/organic electrochemical transistor based on the Physarum polycephalum cell endowed with both sensing and memristive properties
Chem. Sci.
(2015) - et al.
Engineering organic electrochemical transistor (OECT) to be sensitive cell-based biosensor through tuning of channel area
Sens. Actuators A Phys.
(2019) - et al.
In vivo recordings of brain activity using organic transistors
Nat. Commun.
(2013) - et al.
Diffusion driven selectivity in organic electrochemical transistors
Sci. Rep.
(2014) - et al.
Steady-state and transient behavior of organic electrochemical transistors
Adv. Funct. Mater.
(2007) - et al.
Superhydrophobic lab-on-chip measures secretome protonation state and provides a personalized risk assessment of sporadic tumour
NPJ Precis. Oncol.
(2018) - et al.
Organic thin film transistors for large area electronics
Adv. Mater.
(2002) - et al.
Effect of the gate electrode on the response of organic electrochemical transistors
Appl. Phys. Lett.
(2010) - et al.
Electrochemical transistors with ionic liquids for enzymatic sensing
Chem. Commun.
(2010)
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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.