In order to obtain an ink with rheological properties compatible with inkjet printing, silver nanoparticles stabilized powder (AgNPs), with an average size of 100 nm, from NanoTop (China), milli-Q water, ethanol, and Triton ^MT X-100 from Sigma-Aldrich were used. First, 31.58 g of ethanol (96% purity and density 0.8 g/ml) were mixed with 27.62 g of milli-Q quality water in a beaker. 0.8 g of Triton ^MT X-100 was added to the mixture. The solution was stirred at 3,000 rpm with T 25 digital LR disperser, from IKA, while the silver nanoparticles were sequentially incorporated in three steps, 13.33 g every 5 min, to improve homogenization. The stirring was continued for 1 hour, and finally, the solution was sonicated with a Q55 sonicator with standard probe 4422 from Q-Sonicator. The ink preparation was carried out at room temperature, except for the sonication step which was in an ice bath. A homogeneous 40 wt% silver ink was obtained. The ink was tested by drop casting and an Epson TX-135 consumer printer was used to test its printability.

First, 400 μl of ink were deposited on a 25 mm × 25 mm glass substrate using a 100–1,000 μl DragonLab micropipette. Once deposited, the ink was spinned at 1,000 rpm for 30 s with an acceleration of 250 rpm/s. Then, the sample was baked in an ORL-DH oven for 1 h at 150°C. After that, a new layer of silver ink was deposited, repeating the previous process. This process was repeated four times, achieving a ≈ 1 μ m thickness (Supplementary Material). Final annealing was carried out at the same temperature for 20 h, as shown in Figure 1. Next, a TLM (Transmission Line Measurement) structure was fabricated on the silver surface by means of a lithographic process (Supplementary Figure S1A). TLM technique is traditionally used to determine the contact resistance between a metal and a semiconductor Reeves and Harrison (1982), Vinod (2011); however in the context of the present work, TLM was used for measuring the contact resistance between the silver film and the tip of the probe station. For this purpose, an AZ 1500 series photoresin from MicroChemical and a printed mask, with a maximum resolution of 10 μm were used. Supplementary Figure S1B shows the six-point TLM structure obtained, with a width of 1,000 μm and a separation between the centers of the pads of 850, 1,000, 1,250, 1,500 and 1,750 μm, respectively.

An analysis of an ink drop was carried out using a TEM, Brand: JEOL, model 100 CX II, operated at an acceleration voltage of 100 kV and magnifications of 100000x and 270000x to determine the size of the nanoparticles (Supplementary Figure S2). Size measurements were performed with a Digital Micrograph software. The AgNPs size information is useful for estimating the initial capacitance of the ink, which is discussed in a subsequent section. In addition, the annealing process can be observed by means of a SEM throughout the drying time (Figures 1, 4). Supplementary Figure S2 shows that the ink has spherical nanoparticles of 100 nm average size. This information is essential in understanding the ink capacitance behavior at the beginning of drying. In the SEM images, shown in Figure 1, it can be seen that root-like paths are formed during drying. Subsequently, over time the ink turns into a continuous film.

The ink was characterized during the sintering process by impedance measurements performed at constant temperature inside the probe station, using an Agilent E4980A LCR meter and a triaxial thermal-chuck probe station. The samples were dried at 90°C and 120°C, contacted with metallic tips (as shown in Supplementary Figure S4 and the impedance was recorded as a function of time for frequencies from 10 kHz up to 1 MHz.

First, current-voltage (I-V) measurements were carried out on the different segments of the TLM structure. The purpose of these measurements was to determine the contact resistance as well as the resistivity of the ink. Measurements ranged from −40 to 100 mV with a current compliance of 100 mA. (I-V) measurements were carried out in the dark and at room temperature conditions, using a source measuring unit (SMU) Keithley 2636B. In order to determine the thermal coefficient of the ink, this measurement was repeated for different temperatures: 100, 125, 150, and 175°C (Supplementary Figure S3).

Figure 2A shows the (I-V) results of the measurements carried out at room temperature. A linear relationship can be observed for the different segments of the TLM structure. This behavior is expected for the ink once the sintering process has been done since it becomes a resistive element. The measurements show a standard deviation of 0.55%. It can be seen that with the increase in the length of the segment, a smaller slope is obtained, and the resistance increases. The TLM measurement method relates the lengths with the increase in resistivity. The slope of the curve shown in Figure 2A presents units of Ω/mm, which corresponds to the linear resistance of the ink. The intercept is two times the contact resistance between the ink and the tip of the probe station. The slope is 2.5 Ω/mm, and the intercept is 0.17 Ω. Considering the geometric data of TLM structure designed Roberson et al. (2012), it can be concluded that the ink has a resistivity of 19.03 × 10⁻⁸ Ω.m, which is approximately 11.9 times larger than the bulk resistivity of silver. This value is consistent with that reported by different manufacturers for inks being dried at the same temperature SigmaAldrich (2020). As for the behavior of the ink as a function of temperature, it can be seen from Figure 2B that it is also linear; so, it could be modeled by R = R ₀(1 + αT). In this case, α = 7.8 × 10⁻³ Ω/°C, which is approximately twice as large as the silver bulk Reuss (2008). These changes in the intrinsic properties of the material can be explained by considering that sintered inks are porous materials. A percolative model of the sintered ink will be discussed in later sections since the material’s porosity is related to its electrochemical area.

The proposed model can predict the time and temperature the ink must be dried in order to reach a certain impedance. In addition, both the material’s porosity and the time needed for the ink to behave fully resistive could be determined. It is important to note that this is a handy tool for predicting the behavior and usefulness of a custom ink. Once the resistance has been determined, through the percolative model, it is possible to estimate the porosity and the electrochemical area that a given trace of silver will have, which is of great interest for sensors and biosensors. Based on the data obtained in the present work and the information collected from the works Greer and Street (2007) and Samano (2017), an analytic circuit model for ink drying as a function of time is proposed. Most works dealing with ink drying carry out measurements in DC, obtaining an exponential decrease in resistance. This phenomenon can be explained by the diffusive model proposed by Frenkel Frenkel (1945). In Samano (2017), the variation of the ink impedance is analyzed, both in module and phase, as a function of time for different frequencies. From the measurements made in AC, for temperatures of 90°C and 120°C, shown in Figure 3, and in accordance with the behavior observed in Samano (2017), a simplified circuit of a serial RL in parallel with a capacitor is proposed. According to this model, Z = j w L + R 1 − w 2 L C + j w C R . (1)

Figure 3 shows that the fresh ink has a high impedance ranging from a few GΩ to tens of MΩ, depending on the frequency. From this information, it can be concluded that at the beginning of the drying, the ink is mainly capacitive, which agrees with the fact that before the heat treatment, the ink consists of small spheres very close to each other and coated by a dielectric. Before the sintering process starts, the capacitance of each sphere can be estimated as C = r/K Sears and Zemansky (2013), where r is the radius of the sphere, based on the size distribution of Supplementary Figure S2, with a value of 100 nm and K is the Coulomb constant (8.987 × 10⁹ Nm ²/C ²). Each AgNP has a capacitance of C _NP = 1.11 × 10⁻¹⁷ F. Figure 3 shows the experimental results corresponding to two samples that were sintered at different temperatures. It can be seen that the variation of impedance as a function of time is different, but the initial capacitance is the same, proving that the initial capacitance value is independent of the temperature at which the sintering is carried out. Given the geometry of the probe station tips, it is defined that the contact surface is 3.83 × 10⁴ μm ², one-third of the tip area, and the distance between tips is 25 mm. Therefore, considering a layer thickness of 1 μm, a total initial capacitance can be estimated considering each NP as a nanocapacitor, placed in parallel with the contact tip and in series along the trace. The TEM image on the right top of Figure 1 shows that the NPs have a grouping behavior, that can be considered as if one were stuck to the other. Hence, the initial capacitance, C ₀, is given by: C 0 = 2 × C N P × 3.8 × 1 0 6 × 10 2.5 × 1 0 5 = 3.37 × 1 0 - 15 F . (2) This value is consistent with that obtained from the measurements in Figure 3, where it was calculated that the initial capacitance was 3.12 × 10⁻¹⁵ F. This capacitance decreases as the dielectric that formerly coated the nanoparticles evaporates. Since the annealing is done at a temperature lower than the boiling point of the dielectric, the evaporation rate is proportional to the volatility of the substance, which is proportional to the vapor pressure. As shown in Figure 3, the slope of the impedance change depends on the drying temperature; therefore, a β value at 120°C has been experimentally determined. The following expression was obtained for the capacitance: C T , t = C 0 exp − β T t , (3) where β(120°C) = 0.02 1/min. From the analysis of the experimental impedance at the initial time for the different frequencies, an initial resistance R ₀ = 2.09 GΩ was determined, a value in line with previously reported ones (Rama et al., 2015). Based on the Greer model (Greer and Street., 2007), we know that the resistivity has two sintering models, so the following expression was obtained for the resistor: R T , t = R 0 exp − γ T t (4) R T , t = R 1 × ln a + a × 1 − 5.77 t / τ 2 / 5 a − a × 1 − 5.77 t / τ 2 / 5 , (5) where τ = 4.0 × 10⁸ s is the characteristic time for sintering by lattice diffusion Coble (1961), and γ(T)t = 0.2 1/min, as β, is an experimental adjustment parameter, which describes the rate of change of the resistance, therefore depending on the sintering temperature. The fresh ink does not have an inductive characteristic, so L (0, T) = 0. Likewise, the conductive tracks that give rise to inductance, are created at the same rate that the capacitive part disappears. For that reason, we have that: L T , t = L 0 1 − exp − β T t . (6)

Then, the traces begin to join at the same rate as the resistance decreases, resulting in a continuous film. The expression for the inductance is: L T , t = L 0 1 − exp − β T t exp − γ T t − t d , (7) where t _d is a delay and L ₀ = 5 mH.

After replacing Eq. 3, Eq. 4 and Eq. 7 in Eq. 1, the model was simulated in Matlab^®. Figure 4 shows the comparison between the model and the experimental results. After about 30 min, the system becomes fully resistive, and there is no longer a dependency on frequency.

When developing a biosensor, it is essential to know the electrochemical area of the working electrode, which is possible to infer through the material’s porosity. The main application of this model is to determine the impedance that a printed trace will have through its physicochemical properties, the temperature and drying time. Furthermore, when the process of elimination of the capping agent and therefore capacitive behavior is completed, it is possible to infer the porosity of the material through the impedance value. Zhao et al. (2016) describes in his work an exponential model that relates the fraction of the voids in the sample with the resistance value; nevertheless, the proposed model is suitable for resistivities in the same order as the bulk silver resistivity. In the present work, to determine the fraction of the voids the SEM images obtained during the drying process were analyzed following the methodology described by Abdullah and Khairurrijal (2016). The fraction of the voids values obtained for the four SEM images in Figure 4 were 68%, 57%, 49% and 45%, respectively (Supplementary Figure S6). Equation 8 was proposed that fits for fraction of the voids of zero, i.e., a solid silver resistance, up to very high impedance values when the void fraction is up to 70%. R e Z = R 0 ⁡ exp m × f v , (8) where Re(Z) is the real part of the impedance of the sintered ink, R ₀ is the resistance value of a pure silver resistor of the same dimensions, m = 0.3 is the exponential fit coefficient and f _v is fraction of the voids. Supplementary Figure S7 shows a fit with an R ² = 0.9985.