Real-time characterization of hydrogel viscoelastic properties and sol-gel phase transitions using cantilever sensors

Hydrogels are crosslinked polymer networks that contain high water content. The past two decades have seen a sharp rise in fundamental research involving hydrogels and the development of novel hydrogels for various applications in energy storage and biotechnology [1–4]. The need for controlled drug release systems was among the earliest driving forces for hydrogel research in the pharmaceutical sciences [5]. In addition to the ability to incorporate drugs, the ability to incorporate animal cells has led to the widespread use of hydrogels in 3D cell culture and tissue engineering applications [6,7]. For example, a number of studies have shown that the gene expression profiles of cells differ in monolayer tissue culture environments (i.e., 2D substrates) vs 3D matrices, which is often attributed to differences in cell-cell and cell-matrix interactions [6]. Hydrogels have emerged as attractive materials for regenerative medicine applications, including 3D-bioprinted tissues and injectable scaffolds for wound healing and tissue regeneration [8–11]. In combination with their use as energy storage devices (e.g., gel electrolytes for lithium ion batteries) [2,3], passive drug release systems [5,12], tissue scaffolds [13,14], and sensors [15,16], emerging applications to soft actuators, active drug release systems, and complex engineered tissues have driven research on stimuli-responsive hydrogels—for example, in soft robotics and 4D bioprinting fields [17,18]. In such applications, the characterization of hydrogel structure, physical properties, and rheological properties (e.g., crystal structure, dielectric properties, and viscoelastic properties) serve as important indicators of the material’s processability, performance, quality, and response to stimuli. Therefore, identifying new paradigms for the characterization of hydrogels and other gel-based materials is central to accelerating the pace of gel-based materials research and improving the processability and quality of gel-based therapeutics, devices, and other products.

While emerging synthetic techniques such as automated synthesizers and synthetic biology are now being developed to produce materials with unprecedented throughput, characterization loops represent major bottlenecks in accelerated molecular and material discovery workflows [19–21]. The challenges associated with a lack of complementary, high-throughput characterization techniques are also compounded by the breadth of structure and property information that could be useful in assessing material performance and quality across different applications [12,22,23]. In particular, the characterization of hydrogel viscoelastic properties, as well as of other soft materials, presents significant rate-limiting steps because of the requirement for manual sample preparation steps and the time-intensive nature of the tests. The gold-standard instruments for characterization of hydrogel viscoelastic properties are rheometers and dynamic mechanical analyzers [24]. While such instruments are robust and provide reliable information regarding the viscoelastic properties of a sample over a range of strain rates and temperatures, the samples must be manually prepared and measured to high tolerances, and experiments can take upwards of 0.5–1 day per sample depending on the complexity and type of scan being performed. Additionally, such instruments are difficult to integrate with processes, which impedes applications that require viscoelastic property sensing or monitoring. As a result, it remains a pressing challenge to eliminate characterization bottlenecks from accelerated material discovery paradigms. In contrast to traditional characterization methods (e.g., DMA), sensor-based characterization techniques offer measurement advantages associated with sensors, which include process integration through miniaturization and the ability for real-time process monitoring and control. Sensors can also offer improved sensitivity, limit of detection, throughput, and measurement repeatability relative to traditional methods through the use of sensitive miniaturized transducers and the elimination of manual sample preparation steps. Therefore, sensor-based techniques for the characterization of hydrogel viscoelastic properties could provide useful tools for eliminating characterization bottlenecks that currently limit the pace of hydrogel materials research and development.

While various sensors have been created to measure the physical and rheological properties of liquids [25–28], milli- and micro-electromechanical systems have enabled the characterization of viscoelastic properties based on fluid–structure interaction effects [29]. Thickness shear mode (TSM) resonators, such as quartz crystal microbalances (QCMs), were among the first sensors leveraged for viscoelastic property characterization. While TSM resonators enable the characterization of the viscoelastic properties of semi-infinite and thin layers of viscoelastic liquids, viscoelastic properties are obtained using equivalent circuit models, which imposes the requirement of sensor calibration. In addition to shear-mode resonators, dynamic-mode cantilevers have been extensively examined for sensor-based rheological and compositional analysis of liquids, such as viscosity monitoring, chemical sensing, and biosensing [30–35]. Analysis of cantilever sensor response is done using fluid–structure interaction models, which are the same physics that drive the sensor response, in contrast to equivalent circuit models, which are useful for modeling measurements based on impedance responses but are not directly representative of the physical phenomenon. The earliest applications of dynamic-mode cantilevers for characterizing the physical properties of liquids were focused on density monitoring using the well-known inviscid result [36]. In parallel with liquid density monitoring applications, rheological measurements (i.e., viscosity monitoring) were also performed using cantilever sensors by incorporating the frequency- and mode number-dependent hydrodynamic function. An explicit theoretical relationship between cantilever resonant frequency, quality factor, and the viscosity of the surrounding media has been previously reported [37]. Chon et al. validated the theoretical results of the cantilever hydrodynamic function using an atomic force microscope (AFM) cantilever submerged in a range of viscous fluids [38]. Boskovic et al. demonstrated that an AFM cantilever with a known undamped natural frequency (i.e., resonant frequency) and mass per unit length allowed for the simultaneous calculation of the density and viscosity of both gases and liquids [39]. Mather et al. extended the use of cantilever sensors for characterization and monitoring of the viscoelastic properties of liquids through the use of mesoscale piezoelectric cantilevers [29]. In that study, the complex shear modulus was replaced with the viscosity in the previous models developed by Sader [37], Maali et al. [40], and Belmiloud et al. [41], which yielded a system of equations for the determination of the shear storage and loss moduli of a material from the cantilever resonant frequency and quality factor responses (after accounting for the internal damping of the larger cantilever) [29]. While Mather et al. were able to characterize the viscoelastic properties of a polyacrylamide solution, fluid damping effects impeded applications to the characterization of more viscous or viscoelastic materials [29]. Johnson and Mutharasan more recently showed that millimeter-scale cantilevers exhibited a sufficiently high cantilever Reynolds numbers to resonate in highly viscous liquids with dynamic viscosities as high as 10³ cP [42]. These studies suggest that millimeter-scale cantilevers could enable sensor-based characterization of solutions that undergo gelation, materials that exhibit sol-gel phase transitions, and the viscoelastic properties of gel-based materials.

Here, we show that resonance in millimeter cantilevers persists in hydrogels and enables the characterization of hydrogel viscoelastic properties and the continuous monitoring of sol-gel phase transitions (i.e., gelation processes). Sensor signal changes and associated high-frequency viscoelastic properties of alginate and gelatin hydrogels over concentration ranges of 0.25–10 wt. % are compared with low-frequency viscoelastic properties obtained from DMA studies (1 Hz). We obtain the sensor’s dynamic range and limit of detection based on the low-frequency viscoelastic moduli at which the sensor no longer exhibited resonance and the noise level associated with the sensor quality factor response, respectively. Ultimately, this work shows that cantilever sensors provide a complementary platform to traditional DMA for real-time monitoring and viscoelastic characterization of hydrogels and gelation processes and could serve to mitigate existing bottlenecks related to low-throughput characterization loops in accelerated material discovery workflows.

Alginic acid sodium salt, gelatin (300 g bloom from porcine skin), poly(ethylene glycol) diacrylate (PEGDA) (750 Da), 2,2-dimethoxy-2-phenylacetophenone (DMPA), calcium chloride, and ethylenediaminetetraacetic acid (EDTA) were purchased from Millipore Sigma. Lead zirconate titanate (PZT-5A; 72.4 × 72.4 × 0.127 mm³) with nickel electrodes was purchased from Piezosystems (Woburn, MA). Borosilicate glass was purchased from VWR. Glass cylinders and ethanol (200 proof) were from Fisher Scientific. Polyurethane (Fast-Drying) was from Minwax. Epoxy (EA 1C-LV) and cyanoacrylate (409 Super Bonder) were from Loctite.

Composite PEMC sensors with a flush design were fabricated from lead zirconate titanate (PZT) as described in previous studies [see Fig. 1(a)] [43]. Briefly, borosilicate and PZT sheets were diced into chips (2 × 1 × 0.16 mm³ and 5 × 1 × 0.127 mm³, respectively; American Dicing; Liverpool, NY). A borosilicate chip was first bonded symmetrically to one end of the cantilever using cyanoacrylate such that the front of both chips were aligned. Subsequently, 30-guage copper (Cu) wires were soldered to the top and bottom faces of the nickel electrodes on the distal end of the PZT layer opposite to the glass layer. The cantilever was then potted in a glass cylinder (6 mm diameter) with a non-conductive epoxy resulting in a cantilever geometry (3 × 1 × 0.127 mm³). The sensors were then coated with polyurethane via spin coating (1000 rpm for 2 min), which was then allowed to cure at room temperature, to improve adhesion of parylene-c to the sensor. The sensors were then coated with parylene-c (10 μm thick) following vendor protocols (PDS 2010 Labcoter^® 2; Specialty Coating Systems; Indianapolis, IN). Following parylene-c coating, the sensors were annealed for 1 h at 75 °C.

The sensor resonant frequency (f_n), quality factor (Q_n), and phase angle at resonance (ϕ_n), where n indicates the mode number, were continuously monitored with a vector-network analyzer with the impedance option (E5061b-005; Keysight). The sensor’s dynamic mechanical response, here, the frequency response, was obtained via electromechanical coupling effects using electrical impedance analysis, which provides electrical impedance magnitude (|Z|) and phase angle (ϕ) spectra of the piezoelectric layer [|Z| and ϕ vs frequency (f), respectively]. While the absolute amplitude of the cantilever vibration could not be determined without additional measurement techniques (e.g., optical techniques for deflection measurement), which are challenging to implement in hydrogels, the phase angle of the sensor at resonance [ϕ = tan⁻¹(ΔV/I)] provides a measure of the relative displacement in the piezoelectric layer and an indirect measure of cantilever amplitude. Thus, the technique is useful for resonant frequency and quality factor tracking in applications that require analysis in complex fluids and materials that may present challenges to the use of optical techniques. Electrical impedance analysis was performed using a stimulus amplitude of 100 mV AC and zero DC bias across a frequency range (f_n – 10 kHz ≤ f_n ≤ f_n + 10 kHz), which enabled resolution of the off-resonance impedance response and, thus, measurement of the frequency-width-at-half-maximum (FWHM). Sensor signals (f_n, ϕ_n, FWHM, and Q_n = f_n/FWHM) were acquired using a custom MATLAB program based on continuous monitoring of electrical impedance spectra.

Alginate solutions (0.25, 0.5, 0.75, 1, 1.5, and 2.0 wt. %) were prepared by dissolving alginic acid sodium salt in de-ionized water (DIW) at room temperature with continuous stirring. The solutions (5 ml) were chemically crosslinked by depositing a 500 μl droplet of saturated calcium chloride on the surface of the polymer solution approximately 5 mm from the submerged sensor. Gelatin solutions (6, 8, 10, and 12 wt. %) were prepared by dissolving gelatin in DIW at 40 °C with continuous stirring. Following dissolution, the solution was maintained at 40 °C until use. PEGDA hydrogels were prepared by dissolving 1, 2, 3, or 4 g PEGDA in 18.9, 17.9, 16.9, or 15.9 g of DIW at room temperature, respectively, followed by the addition of 0.1 g of 20 wt.  % DMPA in ethanol for final solutions containing 5, 10, 15, and 20 wt. % PEGDA with 0.1 wt. % DMPA. PEGDA hydrogels were cured by exposure to 365 nm UV light for 10 min (1200 μW/cm² at 3 in.; UVGL-58).

The shear storage and loss moduli of the surrounding material, here hydrogels, at the sensor’s resonant frequency (G′_f and G″_f, respectively) were calculated based on previously established fluid-structure interaction models for resonant cantilevers [29,37,39–41,44]. The inertial and dissipative components of the drag force on a vibrating cantilever (g₁ and g₂, respectively) can be written in terms of the frequency-dependent viscoelastic moduli (G′ and G″) as [29]

where a₁, a₂, b₁, and b₂ are Maali’s parameters (a₁ = 1.0553, a₂ = 3.7997, b₁ = 3.8018, b₂ = 2.7364) [40], ω is the angular frequency (here, taken as ω = 2πf_n), ρ is the density of the surrounding material (e.g., fluid), and b is the cantilever width. The components of the drag force can also be calculated from sensor data (i.e., f_n and Q) as [29]

where L is the cantilever length, μ = ρ_cbt is the cantilever mass per unit length, ρ_c and t are the respective cantilever density and thickness, Q₀ and ω_o are the respective quality factor and resonant frequency in the absence of fluid damping (i.e., resonating in vacuum with only internal damping effects present), m_c = ρ_cbtL is the cantilever mass, m_A = ρπb²LΓ′/4 is the added mass, Γ′ is the real part of the hydrodynamic function, and c_i = m_cω_o/Q_o is the internal damping coefficient. Due to the scale of the cantilevers (L = 3 mm), the internal damping was not negligible and was subtracted from the measured value [as described in the term c_i in Eq. (3)]. In calculation of c_i, ω_o, and Q₀ were approximated as ω_o ∼ 2πf_n,air and Q₀∼ Q_n,air, which were reasonable assumptions as discussed in the following sections. The hydrodynamic function was approximated using the relation Γ′ = a₁ + a₂δ/b, where δ = [(2η/(ρω)]^1/2 is the thickness of the thin viscous layer surrounding the cantilever in which the velocity has dropped by a factor of 1/e and η is the viscosity of the fluid. The solution to the system of equations formed by Eqs. (1)–(4) provides the viscoelastic properties of the surrounding material based on cantilever sensor data.

Prior to all experiments, the sensor impedance spectra were characterized in air to obtain the resonant frequency and quality factor. Experiments began by adding 5 ml of the gelatin solution to a 35 mm dish at 40 °C. The gelatin solution was maintained at 40 °C during solution phase studies to prevent gelation. The solution was then cooled to room temperature, which took approximately 5 min. Subsequently, the cantilever was submerged in the solution to a depth that brought the top surface of the polymer solution 30 μm above the cantilever’s anchor. The sensor data acquisition program was subsequently initiated, which enabled continuous monitoring of the sensor signals as the gelatin solution spontaneously underwent a thermoreversible gelation process.

The resonant frequency and quality factor in air (f_n,air and Q_air, respectively) were determined as described in Sec. II F 1. Experiments began by adding 5 ml of room temperature alginate solution to a 35 mm dish. The cantilever was then submerged as described in the previous section and data collection was initiated. Following stabilization of the sensor signals, the alginate solutions were chemically crosslinked by manually applying a 500 μl droplet of saturated calcium chloride to the surface of the solution approximately 5 mm from the anchor of the cantilever. Addition of a 500 μl droplet of DIW served as an in situ negative control. Following the stabilization of sensor signals after chemical gelation, the hydrogels were dissolved by applying 3 ml of a 1 M aqueous solution of EDTA across the surface of the hydrogel.

Characterization of hydrogel low-frequency viscoelastic properties was done using a dynamic mechanical analyzer (Q800; TA Instruments). Cylindrical test specimens of alginate and gelatin hydrogels (diameter = 12.7 mm and thickness = 5 mm) were punched from 5 mm thick hydrogel sheets prepared using the same crosslinking techniques as previously described for the sensor studies. All measurements were acquired by the application of a 15 μm periodic displacement at a constant frequency (1 Hz) and 5 mN preload force in the compression mode. Temperature-dependent data for the thermally responsive gel were acquired under the same conditions using a temperature sweep from 26 to 36 °C at a rate of 0.5 °C/min.

A rheometer (Discovery DH-2; TA Instruments) was implemented with a recessed concentric cylinder geometry. Gelatin solution (8 wt. %) was loaded into the test geometry with a 1 mm gap. Testing conditions of 1% strain and 1 Hz were imposed. The sample was held at 40 °C then quenched to 25 °C at a rate of 5 °C/min, which mimicked the temperature treatment used in the sensor studies. Data collection began when the sample temperature reached 25 °C and continued for 90 min. The time of the gelation process as measured through sensor and rheometer data was normalized by the respective total gelation times. The data were truncated in both sensor and rheometer studies to the point where G′ reached 95% of the maximum (G′₉₅) and subsequently normalized by G′₉₅ for comparison.

As shown in Fig. 1(a), PEMC sensors are actuated and sensed using the same piezoelectric layer, which is referred to as a self-exciting and -sensing design [45]. This design enables the sensor’s mechanical frequency response to be characterized by the electrical impedance response of the insulated piezoelectric layer [photographs shown in Figs. 1(b) and 1(c)]. Thus, electrical impedance analysis enables real-time monitoring of the cantilever resonant frequency (f_n) and quality factor (Q_n). Having previously shown that resonance in PEMC sensors persists in highly viscous liquids of viscosity up to 10³cP [42], here, we examined if resonance in PEMC sensors persists in solutions of gel-forming polymers and resultant hydrogels. Piezoelectric cantilevers enable mechanical frequency response analysis through an electrical measurement technique, specifically, electrical impedance spectroscopy, which is made possible through the electromechanical coupling effects in the piezoelectric layer. Consequently, the amplitude is monitored indirectly through the phase angle of the electrical circuit formed by the sensor materials and equivalent circuit effects of dynamic motion. As shown in Fig. 1(d), the resonance of PEMC sensors is minimally damped by air (f_n,vac = 44.6 kHz and Q_n,vac = 24.8; f_n,air = 44.4 kHz and Q_n,air = 24.7). While millimeter-scale piezoelectric cantilevers exhibit a relatively greater amount of internal damping compared to micro-cantilevers, leading to relatively lower quality factors in vacuum, millimeter cantilevers enable resonant frequency tracking in highly dissipative materials. Considering the dimensions and resonant frequency of millimeter-scale cantilever sensors in a highly viscous liquid [width (b) = 1 mm; f_n ∼ 25 kHz], as well as previously reported rheological properties of gel-forming aqueous polymer solutions (η ≈ 200 cP) [46], the cantilever Reynolds number (Re_c= ρωb²/4η) is ∼150, where ρ is the fluid density, ω = 2πf_n is the angular frequency of the cantilever, and η is the dynamic viscosity of the solution. The result of Re_c > 1 suggests millimeter-scale cantilever sensors should resonate in concentrated solutions of gel-forming polymers. As shown in Fig. 1(d), resonance in PEMC sensors indeed persisted in concentrated solutions of gel-forming polymers (data shown for a 10% gelatin solution).

Having established that PEMC sensors resonate in concentrated solutions of gel-forming polymers, we next examined if resonance persisted in gels following the sol-gel phase transition [see Fig. 2(a)], as opposed to being absent as a result of increased damping effects of a surrounding gel phase. Studies were conducted using gelatin, alginate, and PEGDA hydrogels based on their extensive use across a range of applications, including tissue engineering, food engineering, and bioprinting. As shown in Figs. 2(b)–2(d), the second mode of cantilever sensors exhibited a resonant frequency of 55.4 ± 8.8 kHz and quality factor of 23.8 ± 1.5 in air (n = 5 sensors). The spectral characteristics of the individual sensors are provided in Table S1 of the supplementary material [66]. The second mode was selected because it has previously been shown to persist in high viscosity liquids up to 1000 cP [42]. These values agreed reasonably with the Euler–Bernoulli beam theory and previous finite element studies [47]. As shown in Fig. 2(b), submersion of the sensor in a 6 wt. % gelatin solution caused decreases in the cantilever resonant frequency and quality factor to 33.8 ± 0.3 kHz and 17.4 ± 0.1, respectively. As shown in Figs. 2(c) and 2(d), similar changes in resonant frequency and quality factor were observed upon submersion in 0.25 wt. % alginate and 10 wt. % PEGDA solutions. The shoulder peak near 36 kHz is attributed to a torsional mode that is adjacent to the transverse mode used for tracking and rheological characterization [48]. The peak associated with the torsional mode appears absent in air due to its proximity with the bending mode, making it difficult to observe, as it causes a relatively smaller amount of deformation in the piezoelectric layer at resonance. Upon submersion in solution, the torsional mode becomes observable due to the differences in each mode’s sensitivity to fluid mass loading effects. Corresponding impedance magnitude data over the same frequency range are presented in Fig. S1 of the supplementary material [66]. Importantly, as shown in Figs. 2(b)–2(d), resonance in PEMC sensors persisted in 6 wt. % gelatin, 0.25 wt. % alginate, and 10 wt. % PEGDA hydrogels following crosslinking (i.e., network formation).

Given that applications may use hydrogels across a range of concentrations, we next examined the concentration at which cantilever resonance no longer persisted in the gel phase (i.e., the concentration at which the cantilever quality factor could not be monitored with suitable resolution for characterization and sensing applications). As shown in Fig. 3, we found that resonance persisted in hydrogels over a wide concentration range from dilute polymer solutions up to 15 wt. %. For example, as shown in Figs. 3(a)–3(c), we found that resonance was observable in gelatin, alginate, and PEGDA hydrogels up to 10, 1.5, and 15 wt. %, respectively. These concentrations corresponded to low-frequency storage moduli (E′) of 11.9, 36.2, and 46.2 kPa, respectively, as characterized by traditional DMA studies. Similarly, these concentrations corresponded to complex moduli (⁠$|E∗|$⁠) of 11.9, 37.3, and 47.0 kPa, respectively. This result suggests that the dynamic range and limit of detection regarding sensing of viscoelastic property changes is material dependent. Inspection of Fig. 3 also shows that while gelation caused a decrease in the sensor quality factor for all hydrogels and concentrations thereof examined, network formation in the polymer solution caused a consistent decrease in the quality factor but had varying effects on the resonant frequency.

Having established that cantilever resonant frequency and quality factor could be measured across a range of concentrations in various hydrogel systems, we next characterized the low-frequency viscoelastic moduli (E′ and E″) for each hydrogel examined to establish the sensor’s dynamic range regarding low-frequency viscoelastic moduli sensing. As shown in Table I, a correlation was observed among the cantilever quality factor (Q_n) and the low-frequency viscoelastic moduli (E′ and E″). Overall, the data in Figs. 2 and 3 and Table I show that resonance in PEMC sensors persists in gelatin, alginate, and PEGDA hydrogels across a range of concentrations that have been used extensively in various applications [49,50]. This suggests that PEMC sensors could be used to characterize the viscoelastic properties of hydrogels based on calibration approaches that involve low-frequency DMA measurements as well as potentially enable real-time sensing of hydrogel viscoelastic property changes by continuous tracking of cantilever resonant frequency, phase angle, and quality factor.

Having demonstrated that resonance in PEMC sensors persists in hydrogels formed through differing chemistry and that changes in sensor quality factor correlated with low-frequency viscoelastic moduli, it was next of interest to determine if PEMC sensors enable real-time monitoring of sol-gel phase transitions via continuous tracking of sensor signals. Gelatin solutions undergo a thermoreversible sol-gel phase transition at room temperature without the addition of a curing stimulus, resulting in a gel [51]. As shown in Figs. 4(a)–4(c), gelation of gelatin solutions caused a continuous change in the resonant frequency, phase angle, and quality factor at 6, 8, and 10 wt. % over 1800 s (30 min), which is relatively consistent with previously reported gelation times [52] (data shown for 8 wt. %). At all concentrations examined (6, 8, and 10 wt. %), the resonant frequency underwent an immediate increase before stabilizing after approximately t = 1100 s. As shown in Table I and Fig. 4(a) for the case of 8 wt. %, gelation of 6, 8, and 10 wt. % gelatin solutions caused resonant frequency increases of 1.2 ± 0.8, 1.6 ± 0.3, and 1.5 ± 0.1% (n = 3 studies). In contrast to the resonant frequency increase, which was an immediate effect, both the phase angle and quality factor remained relatively stable for the first 400 s, before ultimately decreasing exponentially until stabilizing at t = 1270 and 960 s, respectively, defining the stabilization time as that at which 95% of the total change in sensor signal had been reached. As shown in Figs. 4(d)–4(f) and Table I, the sensor signal changes upon gelation correlated with the concentration and low-frequency viscoelastic moduli (E′ and E″) of the surrounding hydrogel. As shown in Fig. 4(d), the resonant frequency increased by approximately 1–1.5% upon gelation at all concentrations. The phase angle decreased by 0.1 ± 0.03 and 0.2 ± 0.1% at 8 and 10 wt. % gelatin, respectively (a significant change in phase angle upon gelation of 6 wt. % gelatin solutions was not observed) [see Fig. 4(e)]. As shown in Fig. 4(f), the quality factor decreased by 5.1 ± 0.1, 10.8 ± 2.4, and 18.4 ± 5.6% at 6, 8, and 10 wt. % gelatin, respectively.

Similar to the thermoreversible gelation of gelatin solutions, the chemical gelation of alginate solutions caused exponential decreases in various sensor signals [see Figs. 4(g)–4(i)]. For example, the phase angle and quality factor changed exponentially at all concentrations examined (0.25, 0.5, and 0.75 wt. %; see Figs. 4(h)–4(i)]. The chemical gelation of 0.5 wt. % alginate solutions did not cause a significant change in the resonant frequency but led to an increase in the sensor noise level [see Fig. 4(g)]. A summary of the resonant frequency, phase angle, and quality factor changes caused by the chemical gelation of alginate solutions are summarized in Figs. 4(j)–4(l). Interestingly, while only the quality factor changes correlated with the hydrogel concentration and low-frequency storage modulus across the 6–10 wt. % concentration range for the gelatin system [see Figs. 4(d)–4(f)], both phase angle and quality factor changes correlated with the hydrogel concentration and low-frequency storage modulus across the 0.25–0.75 wt. % concentration range for the alginate system. For example, the phase angle decreased by 0.3 ± 0.1, 0.6 ± 0.1, and 0.9 ± 0.1% as a result of hydrogel crosslinking for 0.25, 0.5, and 0.75 wt. % alginate, respectively [see Fig. 4(h)]. As shown in Fig. 4(i), the quality factor decreased by 9.1 ± 1.4, 22.6 ± 3.7, and 34.2 ± 4.3% during the crosslinking process for 0.25, 0.5, and 0.75 wt. % alginate, respectively. The total resonant frequency changes were not significant relative to the noise level at all concentrations.

These collective results from the gelatin and alginate hydrogel systems show that in addition to enabling real-time monitoring of the gelation process using resonant frequency, phase angle, and quality factor responses, the sensor responses (specifically, the phase angle and quality factor) enabled the quantification of hydrogel polymer content (i.e., concentration) and low-frequency viscoelastic properties across a wide dynamic range through a calibration approach (i.e., a set of offline DMA measurements on experimental standards). Sensor data associated with real-time monitoring of photo-gelation processes are provided in Fig. S2 of the supplementary material [66] for the PEGDA system. The correlations between phase angle and quality factor changes with hydrogel composition and low-frequency viscoelastic properties found in the gelatin, alginate, and PEGDA hydrogels suggest that millimeter cantilever sensors provide useful platforms for real-time monitoring of gelation processes and rheological characterization of sol-gel systems.

Having established a sensor-based approach for the characterization of sol-gel phase transitions and low-frequency hydrogel viscoelastic properties based on benchmarking (i.e., calibration) of total changes in sensor signals against offline DMA data acquired using traditional techniques, we next used a cantilever fluid–structure interaction model for viscoelastic materials [Eqs. (1)–(4)] to examine the behavior of the high-frequency storage and loss moduli obtained at the resonant frequency (G_f′ and G_f″, respectively) and understand their correlation with low-frequency moduli (E′ and E″). In other words, it was our goal to understand the relationship between low-frequency (1 Hz) viscoelastic moduli obtained using traditional characterization platforms (e.g., DMA) and high-frequency viscoelastic moduli (∼35 kHz) obtained using PEMC sensors. Figs. 5(a) and 5(b) show the representative trends in G_f′ and G_f″ during gelation for both the gelatin and alginate systems (data shown for 8 and 0.5 wt. %, respectively). Both G_f′ and G_f″ increased throughout the gelation process, as was observed with low-frequency viscoelastic moduli [53]. However, there was not a crossover point between G_f′ and G_f″, which is typically associated with low-frequency gelation rheology and has been previously reported for gelation of gelatin [54] and alginate hydrogels [53]. Regarding the relative magnitudes of G_f′ and G_f″_,, it is not unreasonable for G′ to be greater than G″ at high frequencies, even in the solution phase due to the relatively slow relaxation time of long biopolymer solutions [55].

As shown in Fig. 5(c), the shear storage moduli of gelatin hydrogels at the resonant frequency (G_f′) were 15 ± 8, 25 ± 0.4, and 31 ± 7 kPa and the shear loss moduli (G_f″) were 14 ± 4, 20 ± 0.7, and 28 ± 4 kPa for 6, 8, and 10 wt. %, respectively. These values are of the same order of magnitude as previously reported values obtained using traditional low-frequency rheological techniques (0.1–10 Hz) [56,57]. For example, Simon et al. found that porcine gelatin exhibited shear storage moduli ranging from 3.2 to 13 kPa over the range 5 to 12 w/v% at 1 Hz [57]. For the case of alginate hydrogels, G_f′ were 47 ± 3, 75 ± 2, and 80 ± 3 kPa and G_f″ were 35 ± 3, 62 ± 9, and 77 ± 3 kPa for 0.25, 0.5, and 0.75 wt. %, respectively [see Fig. 5(d)]. Similar to the characterization of gelatin hydrogels, these values are of the same order of magnitude as previously reported values obtained using traditional low-frequency rheological techniques [58]. Duan et al. found that 2 w/v% alginate exhibited shear storage and loss moduli of 21.1 and 3.4 kPa at 1 Hz. The fact that G_f′ and G_f″ obtained from sensor data were larger than E′ and E″ obtained from DMA is consistent with the frequency response of dynamic moduli, which typically increase with increasing frequency [24]. We note that while storage modulus of a stable system increases with frequency, this is not the case for loss modulus. Thus, the relative increase in moduli is not unexpected when considering the magnitude of the resonant frequency of the sensor (∼35 kHz, see Fig. 2). The limit of detection for changes of G_f′ in gelatin and alginate hydrogels based on sensor quality factor response was 13.2 and 11.4 kPa, respectively [see the associated Q-G_f′ sensor transfer function in Fig. 5(e)]. The limit of detection for changes of E′ in gelatin and alginate hydrogels based on sensor quality factor response was 1.9 and 7.1 kPa [see Table I and Fig. 5(f)]. The data in Fig. 4(k) can also be used to determine the sensor limit of detection based on phase angle response. Based on the DMA results and data in Fig. 4(k), the limit of detection in alginate based on phase angle data was 260 Pa. This greater sensitivity can be attributed to a lower noise level in the phase response data. A discussion of detection limit calculations is provided in the supplementary material [66]. As shown in Figs. 6(a) and 6(b), the shear moduli obtained at resonance (G_f′ and G_f″) exhibited a positive correlation with the low-frequency storage modulus (E′) acquired with DMA. While G_f″ exhibited a positive correlation with alginate hydrogels, the gelatin hydrogel system exhibited limited correlation. This relationship is likely dependent on the hydrogel’s Poisson’s ratio and material property frequency dependence [24]. These results suggest that real-time monitoring of high-frequency viscoelastic moduli using sensor-based approaches provides a promising technique for the characterization of gelation dynamics and quantification of hydrogel viscoelastic properties over a wide frequency range (Hz–kHz).

In addition to correlation between high- and low-frequency viscoelastic properties in cured hydrogels obtained using sensor-based approaches and traditional platforms, we also found that the high- and low-frequency shear moduli exhibited similar temporal responses throughout the gelation process [see Fig. 6(c)]. The time at which the temperature of the gelatin solution reached room temperature after quenching from the solution phase at 40 °C was taken as the time t = 0 in this study. The time scales were normalized by the time at which G′ reached 95% of the total change. It should be noted that the traditional rheology data (1 Hz) exhibit a crossover in G′ and G″ at t = 0.08, while G_f′ was greater than G_f″ for the duration of the experiment based on the cantilever data (34 kHz). While there is no crossover point to indicate a specific gelation time in the PEMC data, it is apparent that the increase in moduli occurred at a relatively later time than in the traditional rheology data. This could be due to the documented effect of increasing shear rate slowing gelatin gelation due to the effects on network formation [59]. As shown by de Carvalho et al., increasing the shear rate from 1 to 1000 Hz not only delayed the onset of moduli increase (often referred to as phase II of gelation) but also depressed the slope of the increase in moduli [59], similar to the data collected using the PEMC sensors. Strain magnitude may also contribute to changes in gelation processes relative to those occurring in the presence of static solid boundaries. For reference, the PEMC sensor data shown here were collected at the cantilever resonant frequency.

Given the previous sections established that PEMC sensors enable the characterization of low- and high-frequency viscoelastic properties and real-time monitoring of gelation processes, we next examined if sensor data could be leveraged to model the dynamics of gelation processes. Sensor responses to chemical gelation of alginate solutions were analyzed using a modified Hill equation based on its previous use in the modeling of gelation processes that were characterized using traditional rheological techniques and is given as [60,61]

where $G′^$ is the normalized storage modulus, t is time, n is the Hill coefficient, and θ is the half-gelation time determined by the time at which G′, and thus, $G′^$⁠, has reached 50% of the total change. The modified Hill equation can also be used to calculate a characteristic gelation rate (P) as

where $Ggel′$ is the storage modulus of the final gel. As shown in Fig. 7(a), the modified Hill model exhibited a reasonable fit to the sensor data (shown for chemical gelation of 0.75 wt. % alginate solutions). Additional analyses of 0.25, 0.5, and 0.75 wt. % alginate hydrogels are shown in Fig. S3 of the supplementary material [66]. The initial time point (t = 0) was taken as the time at which the crosslinking agent was applied to the alginate solution. The dependence of the half-gelation time on hydrogel concentration is shown in Fig. 7(b). Chemical gelation of the 0.25 wt. % alginate solution exhibited the longest half-gelation time of 210 ± 11 s compared to 93 ± 20 and 104 ± 27 s for 0.5 and 0.75 wt. % alginate, respectively. There was no significant difference in the half-gelation time between 0.5 and 0.75 wt. % alginate. As shown in Fig. 7(c), the Hill coefficient and the characteristic gelation rate increased with increasing alginate concentration. The characteristic gelation rates were 564 ± 114, 1844 ± 934, and 3516 ± 944 Pa/s for 0.25, 0.5, and 0.75 wt. % alginate, respectively. The characteristic gelation rate obtained via sensor responses during alginate gelation was higher than previously reported values [62,63]. Junior et al. found a characteristic rate of P = 46.8 Pa/s for the chemical gelation of 2 wt. % alginate hydrogels [63]. The significant increase in the P values extracted here may be largely attributed to the significantly higher final storage modulus measured using resonant PEMC sensors in this study. For example, multiplying the characteristic rate determined using sensor data for 0.75 wt. % alginate (P = 3516 Pa/s) by the ratio of the low-frequency modulus measured by Junior et al. to the high-frequency modulus measured here (G′/G_f′ = 0.92 kPa/80 kPa) yields a calibrated characteristic rate of 40.4 Pa/s, which agrees well with the results of previous studies [63]. These results indicate that in addition to providing quantitative characterization of hydrogel viscoelastic moduli and real-time monitoring of gelation processes, the sensor responses associated with gelation processes can be used for quantitative characterization of gelation process dynamics.

To further evaluate the utility of the sensor for future applications in viscoelastic characterization of hydrogels and continuous monitoring of gelation processes, we next examined the real-time monitoring of hydrogel dissolution processes. As shown in Fig. 8, the application of 3 ml of 1 M EDTA chelating solution following chemical gelation of alginate solutions led to a recovery in the phase angle and quality factor, which is consistent with hydrogel dissolution. EDTA is a well-established chelating agent that is capable of dissolving alginate hydrogels based on its affinity for calcium cations that cause the chemical gelation of alginate solutions [64]. Interestingly, the application of the dissolving agent resulted in a decrease in resonant frequency [see Fig. 8(a); n = 3 repeated studies], which could be attributed to a mass-change response associated with EDTA uptake by the surrounding material. Previous studies that examined the adsorption of metal-EDTA complexes on various surfaces also suggest that EDTA and calcium-EDTA complexes may adsorb to the sensor surface [65]. While the phase angle returned to the original value after the dissolution process [see Fig. 8(b)], the quality factor did not fully recover [see Fig. 8(c)], which may be attributed to a mass-damping effect associated with the observed resonant frequency decreases or differences in the rheological properties of the initial and final solutions. These results support the fact that cantilever sensors can assess a range of sol-gel transition processes, which are of use for characterizing hydrogels across a wide range of applications. The observed change in resonant frequency also suggests that considerations of chemical binding to the sensors during rheological studies, such as binding of polymer or crosslinking agents, may be important for accurate quantification of rheological properties based on sensor data.

In this paper, we report that resonance in cantilever sensors persists in hydrogels. This result was shown to enable the characterization of low- and high-frequency hydrogel viscoelastic properties and the real-time monitoring of sol-gel phase transitions (i.e., gelation processes). Studies were performed on various hydrogel systems that underwent thermoreversible-, chemical-, and photo-gelation processes. Changes in the sensor phase angle, quality factor, and high-frequency shear moduli obtained at the resonant frequency (G′_f and G″_f) correlated with low-frequency moduli obtained using traditional DMA and rheology platforms. These results suggest that real-time monitoring of high-frequency viscoelastic moduli using sensor-based approaches provides a promising technique for characterization of gelation dynamics and quantification of viscoelastic properties of hydrogels over a wide frequency range (Hz–kHz). This work also suggests that cantilever sensors could provide a promising platform for sensor-based characterization of hydrogels that may lead to future breakthroughs in process control and high-throughput characterization. In addition, resonance and quality factor tracking in millimeter-scale cantilever sensors appears to provide an attractive integrated characterization and bioanalytical platform for gel-based biomanufactured products, such as molded or 3D bioprinted hydrogel-based tissues, via real-time detection of rheological property changes, chemical sensing, and bio-sensing.

Z.J.K. and B.N.J. are grateful for the generous support of the National Science Foundation (NSF) (No. CMMI-1739318), which provided funding for the reported work.