2.3.2. Sensor Modelling

An analytical model of the NC membranes soaked with biological solutions and inserted in between the two parallel-plate electrodes is established to provide a physical understanding of the system. In the presence of an electrolyte, the equivalent electrical model considers a double layer capacitance (*Cdl*). The equivalent circuit also considers the NC membrane, modelled through the parallel association of its capacitive (*CNC*) and conductive (*RNC*) properties, as well as the polyester backing, modelled using its capacitive properties only (*CBacking*), since its dielectric losses are considered to be negligible in the studied frequency range. The value of the polymer backing capacitance (~100 pF) is determined experimentally through measurement of its permittivity by means of dielectric measurements (see Section 2.3.1). *CNC* and *RNC*, representing the electrical properties of the NC membranes saturated with saline solutions or bacterial suspensions, strongly depend on the ionic concentration in the solution. The double layer capacitance, representing the interfacial properties is given by [19]:

$$\mathcal{C}\_{dl} = \frac{\varepsilon\_0 \varepsilon\_{r,sol}}{\sqrt{\frac{\varepsilon\_0 \varepsilon\_{r,sol} k\_B T}{2 \cdot q^2 N\_{av} \varepsilon\_{ions} 10^3}}} A\_d \tag{1}$$

with *Cdl* the double layer capacitance, *kB* the Boltzmann constant, *cions* molar ionic concentration of the solution in which the double layer occurs, *<sup>ε</sup>r,sol* the relative permittivity of this solution and *Ae* the surface of the parallel-plate electrode. The values for *Cdl* at different ionic concentrations were simulated using COMSOL Multiphysics v.5.4 (COMSOL AB, Stockholm, Sweden) based on Equation (1), and lie in the 10–100 μF range. Regarding the values of the other parameters, their influence can therefore be neglected at the considered frequencies between 1 kHz and 1 MHz.

The equivalent model has three cut-off frequencies:

$$f\_1 = \frac{\mathbb{C}\_{dl} + \mathbb{C}\_{\text{Backing}}}{2\pi \cdot \mathbb{R}\_{\text{NC}} \cdot \left[\mathbb{C}\_{dl}\mathbb{C}\_{\text{Backing}} + \mathbb{C}\_{\text{NC}} \left(\mathbb{C}\_{dl} + \mathbb{C}\_{\text{Backing}}\right)\right]} \tag{2}$$

$$f2 = \frac{\mathbb{C}\_{dl} + \mathbb{C}\_{Backing}}{2\pi \mathbb{C}\_{dl} \mathbb{C}\_{Backing}}\tag{3}$$

$$f\_3 = \frac{1}{2\pi \cdot R\_{NC}C\_{NC}}\tag{4}$$

#### *2.4. Interdigital Electrodes (IDE) Setup*

2.4.1. Interdigital Electrode Design and Fabrication on Nitrocellulose (NC) Membranes

The deposition of IDE on nitrocellulose substrate was conducted using a physical vapor deposition (PVD) e-gun evaporation technique. Gold IDE were deposited by applying patterned nickel masks on top of the nitrocellulose membrane. This deposition technique is more cumbersome than screen-printing and inkjet printing techniques, which are usually used for deposition of electrodes on paper [31]. However, these have the disadvantage of not allowing precise electrode deposition on chemically untreated nitrocellulose. Hence, PVD is chosen because it allows for precise deposition without inducing variability through additional treatment. The IDE finger width and interdigit gap are 200 μm, which enables the detection of dielectric properties over the whole NC membrane depth (~140 μm) [32].

#### 2.4.2. IDE Impedance Sensing

Figure 2B shows a schematic of the experimental system for IDE measurements. The IDE were connected through toothless crocodile clips to an impedance analyzer (LCR 4284A, Agilent, Santa Clara, CA, USA) through BNC connectors. The impedance spectroscopy measurements were carried out with the LCR, remotely controlled by a computer through the Labview software (Labview National Instrument, Austin, TX, USA) to perform an automatic sweep from 1 kHz to 1 MHz, at voltage amplitude of 20 mV. Before impedance measurement, an open-circuit calibration was performed without any electrical contacts between the crocodile clips. The impedance data were extracted in a magnitude-phase data-structure.

#### 2.4.3. IDE Sensor Modelling

The equivalent circuit of the IDE in Figure 2B incorporates the surficial phenomenon of double layer capacitance through *Cdl* and the volumic phenomena through *Cair*, *Rair*, *CNC*, and *RNC*, corresponding to the upper air layer and lower nitrocellulose layer, respectively. Given the width of IDE fingers, the backing is not taken into account (Section 2.4.1). The

double layer capacitance for IDE electrodes in contact with a given solution is extended from (1) to:

$$\mathcal{C}\_{dl} = \frac{\varepsilon\_0 \varepsilon\_{r,sol}}{\sqrt{\frac{\varepsilon\_0 \varepsilon\_{r,sol} k\_B T}{2 \cdot q^2 N\_{\text{av} \cdot \text{circ}} 10^3}}} A\_v (N - 1) \tag{5}$$

with *Ae* the surface per electrode finger and *N* the number of fingers. The equivalent resistance and capacitance of the nitrocellulose and air volume are given by

$$\begin{array}{ll} \mathsf{R}\_{\mathsf{NC}} = \mathsf{K}\_{\mathsf{cell}} \sigma\_{\mathsf{NC}}^{-1} & \mathsf{R}\_{\mathsf{air}} = \mathsf{K}\_{\mathsf{cell}} \sigma\_{\mathsf{air}}^{-1} \\ \mathsf{C}\_{\mathsf{NC}} = \mathsf{K}\_{\mathsf{cell}}^{-1} \varepsilon\_{0} \varepsilon\_{r,\mathsf{NC}} & \mathsf{C}\_{\mathsf{air}} = \mathsf{K}\_{\mathsf{cell}}^{-1} \varepsilon\_{0} \varepsilon\_{r,\mathsf{air}} \end{array} \tag{6}$$

with *Kcell* the cell constant, *εr* and σ the relative permittivity and conductivity of the sensed nitrocellulose and air volumes, respectively. The cell constant is determined experimentally and incorporates the geometric properties of the IDE [33]. Hence, it does not vary with frequency nor with the electrical properties of the material.

**Figure 2.** Experimental setups and corresponding electrical models investigated towards bacteria detection in this work. (**A**) Parallel-plate probes are a common material dielectric measurement system. The bacterial sample is deposited on the NC membrane and conducted to the test zone by capillarity. A simple electrical model is proposed to consider the dielectric effect of the polyester backing supporting the NC and the electrical double layer that arises from charge redistribution at the interface between the electrolyte and the probe. (**B**) Interdigital electrodes are generally used as sensors to monitor impedance changes at the proximity of the metallic fingers, here deposited on NC membrane. The bacterial samples are deposited on top of the interdigital electrode (IDE)-NC sensor. The model does not include the polyester backing as its impact on the impedance seen by the IDE is negligible due to its depth.

#### *2.5. Sensing of Saline Solutions as Models for Real Water Samples*

Prior to electrical measurements, sodium chloride solutions of various concentrations (*cNaCl*), <sup>10</sup>−5, <sup>10</sup>−4, 5 × <sup>10</sup>−4, <sup>10</sup>−3, 5 × <sup>10</sup>−3, <sup>10</sup>−2, 10−<sup>1</sup> mol/L (M), were prepared by dilutions ofa1M NaCl solution in DI water, in order to model the electrical properties of different types of real water sample and biological buffer (Table 1). Regarding the impedance measurements considered in this study, the parameters of interest to be modelled are the mean ionic strength of the liquid (through adjustment of <sup>c</sup>*NaCl*), and hence its dielectric properties (electrical conductivity and permittivity).

Then, 50 μL of the diluted sodium chloride solutions were then deposited on NC membranes, placed in between the parallel-plate electrodes, and reference dielectric or impedance measurements were performed within 5 min. The parallel-plates were wiped between each measurement to remove remaining water and salt on the electrodes.

**Table 1.** Modelling of the electrical properties at 20 ◦C of real aqueous samples and biological buffers using saline solutions at different concentrations. Mean concentrations and, therefore, dielectric properties are considered as the physico-chemical content of the water samples can vary from place to place [34,35].


#### *2.6. Bacteria Detection in Physiological Buffers*

Before depositing bacterial suspensions, 50 μL of PBS buffer diluted 1000× (PBS:1000) in DI water was deposited on a previously biofunctionalized membrane, and reference dielectric or impedance measurements were performed within 5 min with parallel-plates or IDE, respectively. PBS:1000 was chosen as biological buffer because the largest detection sensitivities were shown to be achieved with low-salt buffer solutions [25], and such lowsalt buffers have electrical properties similar to real water samples. Five min is a time limit before which it is assumed that the wet impedance has not changed by more than 5% due to drying. Then, suspensions of 108, 10<sup>7</sup> and 10<sup>6</sup> CFU mL−<sup>1</sup> of stationary-state *B. thuringiensis* resuspended in PBS 1:1000 were deposited (50 μL) on top of the membrane sample with a micropipette (Figure 2A,B) and spread within the membrane due to capillarity, and impedance or dielectric measurements were performed. In order to observe the global sensitivity of the setup, including both sensitivities of the impedance modulus and phase to bacteria presence, the sensitivities (*S*) in the explored frequency range were computed as the amplitude of the differences in complex impedance measured with and without bacterial cells, in percent:

$$S = \left| \frac{Z\_{\text{Batch}} - Z\_{PBS}}{Z\_{PBS}} \right| \tag{7}$$
