Electrorheological characterization of complex fluids used in electrohydrodynamic processes: Technical issues and challenges

Electrohydrodynamic (EHD) techniques have increased interest from the scientific community since the beginning of the twenty-first century. It works by applying an electric field to the induced fluid ejection from a conductive nozzle onto a substrate [1]. The needs to produce new micro- and nanoscale products has increased to solve challenges present in medicine, electronics, robotics, and additive manufacturing applications [2]. The main EHD techniques are EHD-jet printing, electrospinning, and electrospray (Figure 1a–c). These techniques have been an alternative with low production costs to traditional manufacturing techniques, such as lithography, inkjet printing, screen printing, etc. [1]. The quality of the final products made by EHD techniques depends on a set of parameters, which can be divided into two main groups: operating parameters and fluid parameters [1,2,3]. The operating parameters are the applied voltage, the distance between the needle and the collector plate (working distance), and the flow rate. The fluid parameters are mainly the viscosity, the surface tension, and the electrical properties, which depend strongly on the polymer/particle’s concentration of the fluid and the solvent’s properties. Table 1 shows the threshold values of some properties mentioned above for each EHD technique.

Figure 1

Schematic illustration of the EHD jet printing (a), electrospinning (b), and electrospray (c). Schematic representation of the forces acting on a Taylor cone during an EHD process (d). Adapted from [4,5].

Regardless of the EHD technique used, a Taylor cone will form whenever an electric field is applied to the fluid. This formation occurs as the potential difference generates an electric field around the liquid meniscus at the nozzle tip, bringing the electric charges or ions to the meniscus surface. This process elongates the meniscus and eventually causes it to breakup in the direction of the applied electric field [5]. Once the different forces acting on the meniscus (Figure 1d) reach an equilibrium, Taylor cone is established. The electric forces (Coulomb and dielectric) developed at the meniscus surface are balanced by the surface tension as well as the reverse viscous flow induced in the liquid. Traditionally, the fluid viscosity considered for the dimensionless analysis was measured in the absence of an electric field. However, the impact of an electric field and its strength on the rheological properties, particularly on the viscosity and relaxation time, of fluids used in EHD applications remain unknown. Consequently, it is unclear how these changes will influence the Taylor cone formation process and its stability. Thus, the lack of this knowledge currently prevents us from determining whether some technical issues, such as nozzle clogging, are due to the high particle concentration of the ink, the evaporation of the solvent caused by Joule’s heating from the electric field, or by a sudden increase in the fluid viscosity in the presence of an electric field. Therefore, it is crucial to carefully study the influence of the application of an electric field on the rheological properties of the EHD fluids to optimize the EHD printing processes and avoid technical issues.

Electrorheology is a branch of rheology dealing with the study of the flow and deformation of fluids in the presence of electric field [6]. So far, electrorheology has been focused on the formulation of electrorheological (ER) fluids, known as smart fluids, which can be used in aerospace, robotics, actuators, shock-absorbers, etc. [6,7,8]. During the ER fluid’s formulation, it is necessary to consider several parameters that directly influence the ER effect. The carrier fluid must have a low dielectric constant and electrical conductivity (≲10⁻¹⁰ S/m) [6] and the dispersed phase, that can be either a solid or a liquid material, must have a high dielectric constant, an appropriate electrical conductivity, density, and relatively stable physicochemical properties [8,9]. Dielectric inorganics (e.g., metal oxides, silicate materials) [4,6], conductive organics (e.g., graphene, carbon nanotubes) [4,6] and polymers (e.g., polyaniline, polypyrrole) [10,11,12,13,14,15,16,17], biopolymers (e.g., cellulose derivatives) [6,18], and liquid crystals (e.g., cyanobiphenyls) [19,20,21] are common dispersed materials used to formulate ER fluids [22]. Further, the performance of ER fluids also depends on the particle concentration, size, shape, and geometry of the particles used as dispersed phase [22]. On the other hand, EHD inks can be homogeneous solutions (pure solvents or solubilized material), suspensions, biomolecular inks, and molten materials (polymer and metal) [1]. Typically, these inks must have a low-viscosity and low surface tension, with electrical conductivities ranging from 10⁻⁷ to 10⁻⁴ S/m [23] (Table 1); moreover, the solvent is the main component responsible for the electrical conductivity of the inks, rather than the particles in the ER fluids [6,24,25].

In the EHD processes, ensuring the stable formation of a Taylor cone is crucial for producing high-quality products. Several studies have explored how surface tension, electrical conductivity, and viscosity of inks influence the final products obtained from the EHD processes [1,25,26,27,28,29,30,31]. Focusing on the viscosity parameter, researchers have observed its impact on the diameter of the jet, the diameter of the droplet ejected at the tip of the Taylor cone, and the diameter of the nanofibers produced by electrospinning. Most of the studies reported in the literature successfully integrate the rheological characterization of inks in EHD processes. However, this rheological characterization is limited to examining shear viscosity in order to classify the fluid as Newtonian fluid or viscoelastic and to determine the zero-shear viscosity. This latter parameter helps verify if the hydrodynamic time is greater than the electric relaxation time to ensure a stable formation of a Taylor cone [1,28].

Extensional rheology, which allows the determination of extensional viscosity and the extensional relaxation time, has been largely overlooked in its potential influence on EHD processes. Blanco-Trejo et al. [32] numerically observed significant differences between the Newtonian and viscoelastic fluids in electrospray experiments. They found that small extensional relaxation times of viscoelastic fluids promote an intense extensional deformation of the fluid in the cone-jet transition region. Moreover, the axial polymeric stresses were found to shrink liquid meniscus. These statements were experimentally validated by Yu et al. [29].

In electrospinning, Helgeson et al. [33] highlighted that the finite extensibility limits of polymer solutions stabilize the jet, playing a dominant role in affecting the morphology and diameter of the fibers produced, as the fluid undergoes strong extensional deformation. Later on, Formenti et al. [34] confirmed that the diameter of the nanofibers is intrinsically related to the extensional relaxation time of the fluid, observing that the diameter of the nanofibers decreases as the extensional relaxation time increases [34]. However, a balance between the extensional relaxation time and the shear viscosity is necessary. Viscous polymer solutions with low extensional relaxation time can produce thicker nanofibers compared to those produced from low-viscosity polymer solutions with high extensional relaxation time. Additionally, Akkoyun and Öktem [35] emphasized the importance of establishing predicative quantitative models for electrospinning using rheological models that describe the viscoelastic behavior of the fluids used, which can include polymer solutions, suspensions, and polymer melts.

Despite the established link between rheology and EHD, there is a notable lack of research dedicated to exploring how the presence of an electric field may alter the rheological properties of fluids. Specifically, it remains unclear how the extensional relaxation time of an EHD ink is affected by an electric field. Furthermore, the impact of this effect on the jet diameter, the droplet ejected at the tip of the Taylor cone, the structure of the Taylor cone, and the morphology of the produced nanofiber – whether it has a uniform diameter through its length or forms beads – needs further investigation.

To address these challenges, Rijo and Galindo-Rosales have recently focused on ER characterization of EHD inks composed by 2D nanomaterials dispersed in a leaky dielectric viscoelastic fluid [36,37]. Their studies provided evidence that the electrical conductivity and concentration of nanomaterials affect the ink’s rheology when an external electric field is applied. Additionally, the relative orientation between the electric and the flow fields is another important aspect to consider. Typically, the ER characterization is performed in a rotational rheometer equipped with an ER cell, where the external electric field is oriented perpendicularly to the flow direction; however, the simple shear flow condition is not the dominant flow at any location of the EHD printing technique (Figure 2-top). The ER cell developed by Sadek et al. for capillary breakup extensional electrorheometry (CaBEER) [38] allows for the application of an electric field parallel to the direction of the flow, enabling the study of how the jet thins and breaks in an EHD configuration with greater similarity (Figure 2-bottom).

Figure 2

Electric and flow field configurations occurring in an EHD printing process compared with the configurations allowed by the ER cells developed so far for rotational and extensional rheometers. Reproduced with permission from [36].

The required low viscosity and surface tension values (Table 1), along with an appreciable dielectric constant (≳10) [39] for EHD inks, highlight one of the major limitations of the ER cells commercially available on the market for rotational rheometers, which were designed for larger viscosity values. The aim of this work is to define the limitations of current ER cells and propose methods to mitigate their impact on measurements to avoid inaccurate data collection. Furthermore, the authors discuss how microfluidics devices can bring innovative solutions to overcome these technical limitations on a macroscale.

The rotational rheometers equipped with ER cells have been a key approach to characterize the rheological properties of fluids when an electric field is applied. However, two different ways to create an electric field are commercially available. The MCR 3xx and 5xx rheometers (Anton Paar) and ARES G2 rheometer (TA Instrument) use the same approach to produce an electric field (Figure 3a and b), i.e., the top geometry is in direct contact with an electrified wire connected to an external high voltage power supply. That wire is responsible for additional friction, resulting in artificially higher viscosity values if not considered during the calibration procedure. The rigidity of this wire is affected by Joules’ effect, which is directly proportional to the current circulating through the fluid sample. Therefore, the higher the conductivity of the fluid sample under the same electric field results in a higher Joule effect, reducing the rigidity of the wire.

Figure 3

Electrorheological cells developed by: (a) Anton Paar, (b) TA Instruments, and (c) Bohlin Gemini. Reprinted with permissions of [40].

In contrast, the ER cell of the former Bohlin Gemini CVOR 150 uses an electrolyte solution which is responsible for supplying the voltage generated by the external high-voltage source to the geometry (Figure 3c), avoiding the extra friction effect due to the wire. Peer et al. [40] compared the ER measurements of polyaniline powder dispersed in silicone oil obtained from the ER cells present in Figure 3a and c. The authors concluded that the rheological data obtained by Anton Paar equipped with its wired-ER cell provides higher values of shear viscosity, storage, and loss moduli than those obtained by Bohlin Gemini electrolyte-ER cell. Further, they also noticed that this increment was geometry dependent, as it was more significant for the parallel-plates (PP) than for the concentric cylinders (CC), as depicted in Figure 4.

Figure 4

Shear stress dependent on shear rate: (left) different geometrical arrangements (PP and CC, same rheometer) and (right) same geometrical arrangement (PP or CC, different rheometers). Reprinted with permission of [40].

Moreover, Peer et al. [40] also found that the additional friction effect is negligible when the suspensions have a viscous carrier fluid, such as silicone oil, or a stronger ER effect. However, this does not hold true for suspensions with a weak ER effect, or a viscosity of less than 1 Pa s. Figure 5 shows the shear viscosity measurements of ethyl cellulose (EC) dissolved in toluene for different concentrations of polymer. The presence of the wire in the system influences the viscosity curves regardless of the polymer concentration. Even for toluene, the experimental results obtained can lead to erroneous interpretations of the fluid’s behavior, such as assuming shear-thinning behavior at low and moderate shear rates when in reality, the fluid is Newtonian (Figure 5). The same happens when low-viscosity polymer solutions are used, i.e., the observation of two shear-thinning slopes separated by a plateau of constant viscosity values. These results diverge from the viscosity curves obtained when the fluids are tested under normal conditions for the same range of shear rates, i.e., without the presence of wire in the mechanical system (Figure 5). Even when applying the correction mechanism present in RheoCompass’ software, we propose establishing a new minimum torque value to remove the bad data present for low and moderate shear rates to account for this friction effect. This new torque value is around 30 μN m instead of the 0.1 μN m defined for the MCR 301 rheometer.

Figure 5

Shear viscosity dependent on shear stress for different concentrations of EC dissolved in toluene without application of electric field. Fill and open symbols represent the experimental data obtained from a normal rheological cell (T = 0.1 μN m) and wired ER cell (T = 30 μN m).

Considering the RheoCompass’ correction mechanism and the new minimum torque value, Rijo and Galindo-Rosales [36,37] were able to correctly analyze the ER properties of weakly viscoelastic fluids (Figure 6a); however, Roman et al. [16] did not have this limitation when studying the ER properties of suspension with low-viscosity carrier fluids, thanks to the ER cell design using an electrolyte for applying the voltage to the upper plate (Figure 6b). The suspension studied by Roman et al. [16] consisted of 1 wt% of polyaniline-graphene particles dispersed in silicone oil with a kinematic viscosity of 20 cSt and the suspension studied by Rijo and Galindo-Rosales [37] was 0.25 wt% of molybdenum disulfide (MoS₂) dispersed in 2.5% w/v of EC dissolved in toluene. The test conditions used in the work of Roman et al. [16] were a PP geometry of 40 mm diameter with a gap of 0.5 mm and a PP geometry of 50 mm diameter with a gap of 0.10 mm was used by Rijo and Galindo-Rosales [37]. The minimum shear stress (τ _min) is defined as follows [41]: (1) τ min = 3 T min 2 π R 3 , {\tau }_{\min }=\frac{3{T}_{\min }}{2\pi {R}^{3}}, where T min {T}_{\min } and R R are the minimum torque and radius of the plate, respectively. Assuming a linear velocity profile, the minimum viscosity η _min value will be given by equation (2). (2) η min = 2 H π R 4 T min Ω , {\eta }_{\min }=\frac{2H}{\pi {R}^{4}}\frac{{T}_{\min }}{\text{Ω}}, where H H and Ω are the gap between the plates and the rotational speed, respectively. According to equation (2), for a given fluid, having an ER cell with a higher T min {T}_{\min } , that is the one with the wire, will require the use of larger diameter plate and higher shear rates γ ̇ = Ω R H \left(\phantom{\rule[-0.75em]{}{0ex}},\dot{\gamma }=\frac{\Omega R}{H}\right) to allow reliable measurements. Considering these limitations, it seems that the ER cell developed by Bohlin Gemini is more suitable for studying the influence of the electric field on the rheological properties of low viscosity fluids at shear rates below 100/s, while the ER cell developed by Anton Paar is more suitable for high shear rates.

Figure 6

Quantitative comparison of the flow curves obtained by means of an ER cell using a wire (a) and an electrolyte solution (b). Reprinted with permissions of [16].

The second limitation when using ER cells is related to the maximum electrical current allowed by the voltage generator. In the experimental setup used by García-Morales et al. [42], the power source used in the ARES-G2 rheometer allows the generation of electric fields with a maximum current intensity of 20 mA, which is around 20 times higher than the maximum value allowed by the voltage power supply used in the MCR 301 rheometer used in the experimental setup of Rijo and Galindo-Rosales [36,37]. This limitation means that the ER characterization of fluids is limited by electrical conductivity of the fluid sample. This makes it very difficult or practically impossible to characterize fluids formulated with slightly polar (1-octanol) or polar (water) solvents, and with non-polar solvents laden with electrically conductive particles (graphene) or polymers (polyaniline). This is a major limitation that makes it difficult to rheologically characterize the effect of the electric field on the inks typically used in EHD processes.

Since EHD inks are complex fluids composed of particles, dissolved ionic salts, polymers, surfactants, and solvents, determining their electrical conductivity theoretically can be challenging. In 1873, Maxwell proposed an equation to determine the electrical conductivity ( κ \kappa ) of the suspension as follows [43]: (3) κ k CF = 1 + 3 ( κ p / κ CF − 1 ) φ ( κ p / κ CF ) + 2 − ( κ p / κ CF − 1 ) φ , \frac{\kappa }{{k}_{\text{CF}}}=1+\frac{3({\kappa }_{\text{p}}/{\kappa }_{\text{CF}}-1)\varphi }{({\kappa }_{\text{p}}/{\kappa }_{\text{CF}})+2-({\kappa }_{\text{p}}/{\kappa }_{\text{CF}}-1)\varphi }, where κ p {\kappa }_{\text{p}} and k CF {k}_{\text{CF}} are the electrical conductivities of the particles and the carrier fluid/dispersant medium, respectively, and φ \varphi is the volume fraction of particles. Initially Maxwell’s equation was proposed to determine the electrical conductivity of dilute suspensions with spherical particles. Later, Cruz et al. simplified Maxwell’s equation for concentrated suspensions taking into account the electrical conductivity of the particles and the carrier fluid [44]. (4) κ k CF = 1 − 3 2 φ , for k p ≪ k CF , \frac{\kappa }{{k}_{\text{CF}}}=1-\frac{3}{2}\varphi ,\hspace{1em}\text{for}\hspace{.5em}{k}_{\text{p}}\ll {k}_{\text{CF}}, (5) κ k CF = 1 , for k p = k CF , \frac{\kappa }{{k}_{\text{CF}}}=1,\hspace{1em}\text{ for}\hspace{.5em}{k}_{\text{p}}={k}_{\text{CF}}, (6) κ k CF = 1 + 3 2 φ , for k p ≫ k CF . \frac{\kappa }{{k}_{\text{CF}}}=1+\frac{3}{2}\varphi ,\hspace{1em}\text{for}\hspace{.5em}{k}_{\text{p}}\gg {k}_{\text{CF}}.

With the emergence of electrorheology, Whittle et al. [45] studied the dependence of the ER response/effect of a fluid on its electrical conductivity and dielectric properties. They concluded that the electrical conductivity of ER fluid is determined by (7) κ = φ k p + ( 1 − φ ) k CF . \kappa =\varphi {k}_{\text{p}}+(1-\varphi ){k}_{\text{CF}}.

As mentioned in Section 1, the ER fluids are typically made up of dielectric solvents with κ < 10⁻¹⁰ S/m. According to Whittle et al. [45] the conductivity of the particles strongly determines the conductivity of the ER fluid. According to equation (7), the electrical conductivity of the ER fluid depends primarily on the first term of the equation, as demonstrated by Liu et al. [46] and Dong et al. [47].

For EHD inks, the solvent or carrier fluid significantly influences the ink’s electrical conductivity. Regardless of the formula used to calculate the fluid’s electrical conductivity, whether ER or not, this property is assumed isotropic and independent on the electric field strength or the deformation rate imposed on the fluid, whether it is shear or extensional. Understanding how the electrorheology cells available on the market for rotational rheometers (Figure 3) work, these systems can be translated into an electrical circuit (Figure 7), where R plate, top {R}_{\text{plate,}\text{top}} and R plate, bottom {R}_{\text{plate,}\text{bottom}} represent the electrical resistance of the plates used in the ER cell and R fluid {R}_{\text{fluid}} is the electrical resistance of the fluid.

Figure 7

Schematic illustration of the electrical circuit represents the way an ER cell works.

Stainless steel is the material used by rotational rheometer manufacturers to produce geometries that allow the creation of electric fields, which means that its electrical conductivity is several orders of magnitude higher than the electrical conductivity of the fluid to be characterized electrorheologically. This huge disparity in electrical conductivity between the parallel plate geometry and the fluid means that the electrical resistance induced in the system by the plates can be neglected. Considering that electrical conductivity is an isotropic property and Ohm’s law is respected ( U = R fluid I U=\hspace{.25em}{R}_{\text{fluid}}I ) when a voltage ( U U ) and current ( I I ) are applied to the system [48]. The electrical resistance of the fluid ( R fluid {R}_{\text{fluid}} ) filling the gap of a parallel plate geometry can be calculated as follows [48]: (8) R fluid = ρ e H A , {R}_{\text{fluid}}=\hspace{.25em}\frac{{\rho }_{\text{e}}H}{A}, where H H and A A are the gaps between the plates and area of the plate, respectively, and ρ e {\rho }_{\text{e}} is the electrical resistivity of the fluid which is the inverse of the electrical conductivity ( ρ e = 1 / κ {\rho }_{\text{e}}=1/\kappa ). Combining Ohm’s law and equation (8), it is possible to determine the maximum conductivity that can be measured for a given voltage generator (equation (9)): (9) κ max = H π R 2 I U = 1 π R 2 I max E , {\kappa }_{\max }=\frac{H}{\pi {R}^{2}}\frac{I}{U}=\frac{1}{\pi {R}^{2}}\frac{{I}_{\max }}{E}, where E = U H E=\frac{U}{H} is the electric field imposed on the fluid and I max {I}_{\text{max}} is the maximum electrical current allowed by the voltage power source. Figure 8 shows the relative importance of the diameter of the plate, the imposed electric field and the maximum intensity of the voltage generator in order to allow the ER characterization of common solvents used in EHD printing process. It becomes evident that having a voltage generator allowing higher current intensity values is paramount for characterizing higher conductivity fluids. Figure 9 shows the best option for the maximum current intensity of a high-voltage power supply (HVS) to be coupled to rotational rheometers. For a current intensity of 200 mA, it would be possible to carry out the ER characterization of slightly polar solvents used in EHD techniques.

Figure 8

Comparison of the maximum conductivity allowed for each electric field and plate radius for the voltage generator commercialized by Anton Paar (a) and TA Instruments (b).

Figure 9

The dependence of the electrical conductivity on the plate radius of a rotational rheometer coupled hypothetically to an HVS that allows a maximum current intensity of 200 mA.

The studies conducted by Liu et al. [46], Dong et al. [47], García-Morales et al. [42], and Rijo and Galindo-Rosales [36] included measurements of the electrical conductivities of their fluids prior to ER characterization. These studies show that the electrical conductivity values fall within the acceptable range for ER characterization, with electric fields strength between 1 and 10 kV/mm. This range depends on the size of the plate used and the maximum current intensity allowed by the high-voltage power source (Figure 8). These results confirm the validity of the assumptions made for deriving equation (9).

In characterizing the viscoelastic behavior of these fluids under oscillatory shear flows, the same issue regarding the fluid sample’s conductivity limit persists. The increased minimum torque caused by the presence of the wire makes it challenging to determine the linear viscoelastic regime from an oscillatory strain-amplitude sweep. This is because the minimum value of any viscoelastic modulus is proportional to the low-torque threshold, defined as, G min = F τ T min γ 0 {G}_{\min }=\frac{{F}_{\tau }{T}_{\min }}{{\gamma }_{0}} [49], Where, F τ {F}_{\tau } is a geometric proportionality factor relating the simple shear stress ( τ 12 {\tau }_{12} ) to the torque ( T T ) and γ 0 {\gamma }_{0} the strain oscillation amplitude. Raising the low-torque limit by two orders of magnitudes would require a severe reduction in F τ {F}_{\tau } , effectively preventing viscoelastic characterization of EHD fluids in oscillatory shear tests under the presence of an external electric field. Furthermore, the low viscosity of EHD fluids imposes significant constraints on fluid inertia during frequency sweep experiments. Consequently, characterizing the viscoelastic properties of EHD samples in the presence of an external electric field must rely on determining the relaxation time under extensional flow.

Many manufacturing processes, including fiber spinning, blow molding, spray coating, electrospinning, electrospray and EHD-jet printing [4,50,51], are governed by elongational deformations. Hence, understanding the extensional properties of the fluids used in these processes is of paramount importance. In an effort to understand the effect of the electric field on extensional rheological properties, Sadek et al. [38] pioneered the development of an ER cell that can be used in the capillary break-up extensional rheometer (CaBER). The authors reported that the extensional behavior of the ER fluids consisting of a dispersion of cornstarch in olive oil depends on the particle concentration and the applied voltage; moreover, they confirmed that the Hencky strain is another relevant parameter in the characterization of ER fluids [38]. García-Ortiz et al. [52] further developed the study analyzing the influence of the polarity of the electric field during the filament thinning process for Newtonian and ER fluids. They observed that the filament lives longer when polarity is against gravity. Later, Rubio et al. [53] measured the filament electrical conductivity and extensional relaxation time for polyethylene oxide dissolved in deionized (DI) water and in mixture of glycerol and DI water. The authors reported some limitations during the experimental campaign. One was that the HVS could not ensure a constant electric field strength due to the electrical conductivity of the fluids. To overcome this limitation, the authors proposed to use a system of resistors in series between the fluid and the HVS [53], as shown schematically in Figure 10. However, this is not the best option to solve the problem, because the true voltage applied to the fluid is a little lower than the voltage applied to the whole system.

Figure 10

Schematic solution proposed by Rubio et al. [53]. Reprinted with permission of [53].

The diameters of the plates typically used in CaBER are 4, 6, and 8 mm. Applying the approach followed in Section 2 to determine the maximum current value that allows us to characterize the ER properties of slightly polar solvents under extension flow, Figure 11 shows that a voltage source with a maximum current intensity of 20 mA is sufficient to overcome the limitations present when a voltage source of 1 mA is used. In addition, the analysis shown in Figure 11 for a 20 mA voltage source makes it possible to study the ER properties of DI water for a range of electric field strengths between 0.1 and 5 kV/mm for a 4 mm diameter plate without the need to manufacture a smaller plate for the CaBER. This analysis is consistent with the experimental work done by Rubio et al. [53], where the authors were able to study the electrical and rheological properties of aqueous polyethylene oxide solutions when subjected to strain stresses using a CaBER coupled to a HVS of 20 mA. However, associated heat and mass transfer problems would be expected, particularly at the later stages of the thinning process, when the cross-sectional area of the filament is minimal.

Figure 11

The dependence of the electrical conductivity on the plate diameter of a CaBER coupled to a HVS of 1 mA (left) and coupled hypothetically to HVS of 20 mA (right).

In spite of this limitation, Rubio et al. [53] could study the influence of the electrical conductivity with the minimum filament diameter during the thinning process. However, they could not ensure if the electrical conductivity depends on the electric field strength or there were other effects that could be present, such as Joule’s effect [54]. To see this effect, two different experimental methods can be used: (i) use a color camera coupled with thermal lenses, and (ii) use the Schlieren technique [55]. In the first option, it is possible to observe the temperature gradient inside the filament during the thinning process. The second option consists of visualizing density variations in transparent media [55]. If the Joule’s effect is present, the Schlieren images allow us to see the density variations promoted by the fluid’s evaporation. Nevertheless, these techniques would require further experiments for testing and validation, potentially using the CaBEER.

In last decade, Dinic et al. [56] proposed a new experimental method, known as dripping-onto-substrate (DOS) extensional rheometry, for determining the relaxation time and the extensional viscosity for low-viscosity elastic liquids, beyond the range measurable in the standard geometries used in the CaBER device. This methodology may, in principle, face the same problems as CaBER regarding partial evaporation; however, Robertson and Calabrese [57] successfully developed a chamber to enclose the sample in an environment saturated with solvent vapor in order to mitigate evaporation during DOS extensional rheology measurements. Rubio et al. [58] implemented an electrified version of this DoS methodology (Figure 12) to study experimentally and numerically the thinning of Newtonian leaky-dielectric filaments subjected to an axial electric field; although they considered moderately viscous liquids with high electrical permittivity, which are far from EHD printing conditions, they confirmed that the electric force delays the free surface pinching due to polarization stress. In their work, the partial evaporation of the solvent was not an issue, and the effect of viscoelasticity was not considered.

Figure 12

Electrified version of the drop-on-substrate methodology for performing electrorheology. Reprinted from [58,59,60,61].

Rijo and Galindo-Rosales [36,37] studied the extensional rheological properties of nonpolar viscoelastic inks laden with 2D nanoparticles in the CaBEER. They noted the need to use the slow retraction method developed by Campo-Deaño and Clasen [62] to minimize the inertial effects present in the fluid during the thinning process. Despite the high volatility of the solvent, partial evaporation of the fluid during the experiment was discarded due to the low electrical conductivity and the short experiment time. However, Rijo and Galindo-Rosales [36,37] observed that the 2D nanoparticle suffered from migration and the EC providing elasticity to the carrier fluid induced the formation of vortices inside the liquid bridge under the influence of an electric field. The 2D nanoparticle migration affects the stability and homogeneity of the fluid under study during the experiment timescale. Moreover, the presence of recirculation may jeopardize the condition of uniaxial elongation flow due to the presence of local shear rates, limiting the usefulness of the study to a qualitative comparison of the apparent relaxation time. However, it has been confirmed that the CaBEER allows to replicate very well the conditions occurring in the EHD printing process, where the consistency and homogeneity of the ink formulated with colloidal particles is “disrupted during the ink jet formation and regions with low and high concentrations are created” [63]; additionally tangential electrical stresses acting on the liquid–gas interface in the CaBEER experiments induce the vortex formation, as in the Taylor’s cone during the EHD printing processes [64]. Furthermore, the shape of end-drops at the end of the CaBEER experiments allows to assess if a formulation will be able to form the Taylor’s cone in EHD printing process [37,65].

Microfluidics is the science and technology of systems that process or manipulate very small amounts of fluids in geometries with characteristic length scales below 1 mm [66]. The small length scales in microfluidics enable the generation of flows with high deformation rates while maintaining a low Reynolds number (Re). These unique flow characteristics promote strong viscoelastic effects, quantified by the elasticity number (El), which scales inversely with the square of the characteristic length (L) [67]. As the scale is reduced, these effects are enhanced. These distinctive flow properties, combined with numerical optimization techniques [68,69,70,71,72,73] provide a rich platform for rheologists to conduct rheometric investigations of non-Newtonian flow phenomena at small scales. This approach overcomes the limitations of macroscopic rheometry [74,75,76,77] and allows exploration of regions in the Wi–Re map unreachable by commercial rheometers [78].

Additionally, a microfluidic-based rheometer-on-a-chip offers several practical advantages, such as low volume samples, minimizing partial evaporation of solvents and the ability to serve as an online rheological sensor in various industrial processes. The VROC^® Technology [79], commercialized by Rheosense [80], is currently the only device allowing the measurement of steady shear flow curves and can even be used to characterize complex fluids under extensional flows [81]. Until 2023, Formulaction commercialized the FluidicamRheo^TM, a shear microrheometer based on the technology developed by Colin and co-workers [82]. This device is an optical microfluidic viscometer where viscosity determination does not rely on pressure transducers [83]. Other approaches and designs of microrheometers exist [84,85], but none of them have reached the market yet nor can perform electrorheology. Therefore, to the best of authors’ knowledge, it is currently not possible to perform shear electrorheometry at the microscale.

The difficulty in providing homogeneous electric field throughout the whole volume of the fluid sample might have discouraged the exploration of electrorheometry at microscale. Recently, Galindo-Rosales et al. [86] disclosed an invention to perform microelectrorheometry, either shear or extensional. As depicted in Figure 13, there is a main microchannel through which the fluid sample flows, driven by either a syringe or a pressure pump, and two auxiliary microchannels, filled with a metal or a metal alloy, that are parallel to each other, either arranged parallel or perpendicular to the main microchannel. These auxiliary microchannels operate as electrodes for generating an electrical field to be applied to the fluid sample. With this configuration, it is possible to impose an external electric field parallel or perpendicular to the fluid flow, which, depending on the chosen geometry, can be designed for shear or extensional flow. This proposal ensures the homogeneity of the electric field within the zone of analysis, and it would be compatible with the VROC and FluidicamRheo devices; however, they have not yet been tested experimentally.

Figure 13

The four possible configurations for the microelectrorheometer allowing to expand the limitations of the current ER cells available for the commercial rheometers. Different configurations for the integrated Microelectrorheometer, all of them containing the following components: the main microchannel (1) through which the fluid sample flows in the direction indicated by (24), the external electric field (4) between the auxiliary microchannels (2) acting as electrodes by means of the voltage supplier (3). Reproduced from [87].

Nevertheless, the study and development of microfluidic devices for electric field applications has been well-documented in the literature. In these devices, the electrodes may or may not be aligned with each other and/or with the flow direction, and have been used to study electrophoresis, electroosmosis, and other electrochemical phenomena [87,88,89,90,91]. Numerous research works have reported that the presence of an electric field can help control droplet size, particle focusing, and separation, material synthesis or fluid pumping [87,88,89,90,91]. Recently, Singh et al. [92] studied the possibility of producing microdroplets through an EHD process in a T-junction microfluidic device. The publication of these works in literature provides rheologists with hope and inspiration for the development of microelectrorheometers, considering the different manufacturing methods of microfluidic devices that allow the application of electric fields.

The electrical conductivity of the fluids used in EHD processes often relegate electrorheology to a secondary role. This lack of integration means that it is challenging to predict the effects of viscosity and relaxation time in the presence of an electric field when the fluid undergoes elongational or shear deformation. Knowing these fluid properties beforehand can help establish relationships between operating and fluid parameters, preventing clogging phenomena, and optimizing EHD processes. For example, understanding beads-on-a-string formation can indicate whether nanofibers from electrospinning replicate this effect, or if satellite droplets would be generated during main droplet detachment in EHD-jet printing.

To reinforce the relationship between electrorheology and EHD, the authors suggest using the most suitable plate diameter and HVS for shear rheological characterization of moderately conductive fluids, ensuring the highest benefit-cost ratio. For studying the relaxation time and the formation of beads-on-a-string, 4 mm plates and a 20 mA HVS are the most suitable tools for CaBER.

Further research on microelectrorheolgy can open new avenues for enhancing the ER characterization of EHD inks. This advancement will enable the refinement of ink formulations and the optimization of the EHD printing processes.

A proper ER characterization of EHD fluids is paramount for the development of cost-effective and robust formulations that can withstand the demanding conditions of EHD processes. Furthermore, it will also bring one significant opportunity in the development of more sophisticated models that can accurately predict the behavior of these fluids under varying electric fields and shear conditions. This could lead to more precise control over processes like electrospinning and EHD-jet printing, improving the quality and consistency of the produced materials. In conclusion, while significant progress has been made, research and innovation in electrorheology are vital to further develop EHD processes.