An experimental and computational study is presented on the interfacial dynamics of a colloidal fluid having both high electric conductivity and high magnetic permeability in the presence of simultaneous electric and magnetic stresses on the fluid/air interface. A transient computational model is developed that simultaneously solves the Navier-Stokes equation and Maxwells’ static equations to predict the transient geometry of the fluid subject to electric and magnetic stresses. This model is first applied to predict the onset of spray emission from a capillary needle electrospray source subjected to a magnetic field. The experimentally determined onset of emissions at each magnetic field agreed well with those predicted by the simulation tool. The predictive modeling tool was then applied to analyze the interfacial profile of a sessile droplet subjected to both electric and magnetic fields. The model captured the geometric evolution of the droplet for voltages up to approximately 85% of the critical onset voltage; near the onset, the model slightly overpredicted the droplet deformation. Using the interfacial stress obtained from the modeling tool, a quantitative discussion is made regarding the roles and magnitudes of the electric and magnetic stress components on the lead-up to the emission instability.

## I. INTRODUCTION

The deformation of a fluid meniscus in an electric field, which in the case of sufficiently strong fields results in emission of charged droplets, has many practical applications including mass spectrometry, pharmaceutical production, nano-fabrication, and spacecraft propulsion. Traditionally, this phenomenon of emission is achieved only in the presence of an electric field with a magnetically neutral fluid and is thus termed “electrospray.” However, in the case of a magnetic fluid, it has been demonstrated that emission can be assisted through the use of a magnetic field, changing the dynamics of the fluid meniscus and emission.^{1,2} The aim of this research was to develop a modeling tool to predict the deformation of a fluid interface under simultaneous electric and magnetic stresses and to analyze the dynamics leading up to spray emission.

When a uniform electric field is applied to a free droplet that has a closed bounding interface with another fluid (or vacuum), the droplet elongates in the direction of the applied field. At equilibrium, the droplet assumes the shape of a prolate or oblate spheroid^{3,4} with geometry dependent on the coefficient of surface tension, the permittivity and conductivity of the fluid relative to that of the surrounding, and the volume of the fluid droplet. Likewise, when a fluid droplet of high permeability is subject to a magnetic field in a similar fashion, the fluid droplet will similarly stretch in the direction of the applied field. This phenomenon has been investigated in great depth utilizing analytical methods^{5–10} for the presence of a single field (either E or H) with much less attention given to the case when both fields are present.^{11,12} When the fluid droplet is pinned to a surface, the shape of the droplet is complicated by the addition of the contact surface energy. Previous investigation into the deformation of sessile droplets under the influence of either the electric or magnetic field has been studied numerically,^{13–18} but little attention has been given to the meniscus behavior of a sessile droplet under the combined action of electric and magnetic fields.

For strong electric fields, the fluid meniscus will deform into a sharp point until a threshold is reached at which time the pointed meniscus begins to emit a jet or even individual molecular ions. The precise voltage needed to achieve emission is referred to as the onset voltage, *V*_{c}. Several models have been developed to predict the onset voltage of electrospray (with no magnetic stress). The most eminent of these models is by Prewett and Mair. In this model, the onset voltage for electrospray is predicted as the condition when the apex electric stress equals the capillary surface stress; assuming the meniscus is a paraboloid having tip radius *r* separated from the extractor electrode by a distance *d*, the Prewett and Mair approximation becomes^{19}

While this relationship is intuitively straightforward, it is only an approximation and is not physically rigorous. The premise of Prewett and Mair is that the fluid will emit spray when the electric traction on the interface exceeds the surface tension. In reality, this condition will never occur; the capillary stress depends upon the coefficient of surface tension *and* the interface radius of curvature. In response to increasing electric stress, the pliant meniscus will sharpen without limit (radius of curvature *r* can decrease to zero) and thus there is no practical upper bound on the capillary stress that can be “exceeded” by the electric stress. Despite this, for many instances, the Prewett and Mair relationship agrees adequately with observations in select cases—typically when some external structure such as a hollow capillary or externally wetted needle is present to impose a geometric curvature length scale independent of the fluid properties. Note that magnetic stresses are not included or anticipated in the onset model of Prewett and Mair. Another electric-field-only onset model was developed by Krpoun and Shea.^{20} This approach modeled the fluid deformation leading up to onset using a combined numerical model for the stresses and an analytical model for the geometry. Unlike the Prewett and Mair approach, the technique of Krpoun and Shea includes a deformable meniscus; however, the meniscus shape is constrained to be a conic section defined by a Bernstein-Bezier curve, and thus the self-consistent meniscus profile is not obtained in the process of solution.

The work reported here was motivated by a desire to understand the onset conditions for a new type of electrospray, where both electric and magnetic fields are applied to a superparamagnetic fluid to achieve spray. King *et al.*^{1} developed a highly conducting superparamagnetic colloid, termed as an ionic liquid ferrofluid (ILFF), that responds strongly to both magnetic and electric stresses. When a magnetic field is applied to a small pool or droplet of the ILFF sharp tips form via the normal-field (Rosensweig) instability. Subsequent application of an electric field results in spray from each of the tips.^{21} In this technique, the fluid is neither coated on the external surfaces of a solid needle nor fed through a hollow capillary, and thus there is no imposed geometric length scale for the meniscus curvature. Prediction of the onset voltage for magnetically augmented ILFF electrospray is complicated by two factors: (1) both the electric and magnetic stresses depend strongly on the shape of the meniscus (which in turn depends on the applied fields), and (2) the fluid free surface deforms gradually under the applied fields such that there is no relevant “tip diameter,” such as that would be present in the case of a hollow-capillary electrode that can be used to estimate the onset electric field.

## II. OVERVIEW OF RESEARCH

The overarching goal of our research was to understand the interrelation between magnetic, electric, and surface tension stresses during the run-up to spraying instability of an ILFF. We approach the problem by observing the onset of instability through laboratory testing and also analyzing the laboratory configurations using a dynamic fluid/electromagnetic simulation. Two configurations were analyzed: the first configuration was the meniscus formed at the tip of a hollow capillary needle, and the second configuration was a sessile droplet on a flat plate. Both configurations were subjected to controlled electric and magnetic fields introduced via biased electrodes and a Helmholtz coil.

## III. FLUID ELECTROMAGNETIC STRESS TENSOR

The stress balance on a fluid-fluid interface can be expressed as follows:

where **T** is the fluid stress tensor, *σ* is the surface tension between the two fluids, and $\u25bf$_{t} denotes the operation along the tangential plane of the fluid interface. For a simple fluid, the stress tensor is dependent only on the local fluid pressure and the fluid velocity. The resulting stress tensor can be expressed as

In the presence of magnetic and electric fields, the Maxwell stress tensor must be included in addition to the terms in Eq. (3). For the case of an incompressible, nonlinear magnetic material, the magnetic stress tensor becomes^{21}

For an incompressible, linear polarizable material, the electric stress tensor is^{22}

The body forces on a fluid volume depend on the gradient divergence of the fluid tensors

In the case of an isothermal, incompressible colloidal fluid, with no charge density or gradient in the particle concentration, the net body force is zero. Consequently, magnetic body forces are not present in this investigation.

## IV. IONIC LIQUID FERROFLUID

The ferrofluid used for all work reported here was a colloid created by stabilizing 17.6% w/w iron nanoparticles in EMIM-NTf2 ionic liquid. The ILFF colloid was prepared by Dr. Brian Hawkett and Dr. Nirmesh Jain at the University of Sydney. The preparation process of the ILFF has been reported previously.^{1}

The surface tension and density of the fluid were measured, and the magnetic M-H relationship was characterized in order to provide inputs to the simulation tool. Using a Kruss K100 force tensiometer and the “plate method,” the surface tension was determined to be 32.389 ± 0.070 mN/m. The density was measured to be 1.815 g/cc.

The magnetization response of the ferrofluid was characterized by the University of Sydney using a vibrating sample magnetometer (VSM). The magnetization relationship for an ideal ferrofluid is often expressed as a Langevin function,^{21}

where *τH* is the ratio of the magnetic to kinetic energy of each nanoparticle and *β* is a function of the bulk magnetization of the nanoparticle material and the volume fraction of the particles in the ferrofluid. The relative permeability as a function of *H* thus becomes

Figure 1 shows the Langevin function fit to the M-H curve measured using the VSM. This curve was then used to specify the magnetic response of the fluid in numerical simulations.

## V. CAPILLARY NEEDLE ELECTROSPRAY EMITTER

Electrospray from a hollow capillary needle has been studied at great depth in the literature. In order to provide a baseline comparison, the first objective was to observe how the addition of magnetic stress changes the well documented onset of capillary electrospray. The ILFF was contained within a capillary needle biased to a high voltage with a grounded electrode in proximity. The needle fixed the base of the fluid meniscus, making the contact line independent of the strength of any applied field. This geometry enables a direct comparison between the onset with and without a magnetic field. A direct comparison cannot be obtained with a sessile droplet.

### A. Experimental setup

The needle emitter, shown in Fig. 2, was positioned within a Helmholtz coil capable of generating a magnetic field of 200 G. The emission source was insulated from the heat generated by the coil through the use of multi-layer insulation. A glass vial, containing a fluid reservoir, was connected to the emission source via a continuous glass capillary. This vial was positioned above the needle tip such that hydrostatic pressure would form a meniscus at the needle exit. The fluid was biased with respect to a grounded extraction electrode. Emission current was measured using an ammeter placed between the power supply and the emitter. The needle and extraction electrode were separated by 1.32 mm, and the needle’s inner diameter was 75 *μ*m. This study utilized the same fluid used in the sessile droplet study described in Secs. IV and V.

### B. Computational approach

The coupled electromagnetics and fluid mechanics of the capillary needle experiment were modeled using COMSOL. The computational domain for this study is presented in Fig. 3. The two-phase flow interface was applied to region 1; outside this region, fluid motion will be minimal and will have negligible influence on interfacial dynamics. This region was therefore treated as a static medium to reduce the computational load. Inflow and outflow boundary conditions were included in region 1 to permit the ILFF to enter and displaced air to leave the region. The electrostatics and magnetostatics interfaces were applied to all domains.

The fluid deformation was modeled using the two-phase flow physics module. This module is based on the arbitrary Lagrangian-Eulerian technique—a method which allows interfacial nodes to move to accurately model the ferrofluid-air interface while the interior nodes move to optimize element geometry. This physics interface modeled the fluid using an incompressible laminar flow version of the Navier-Stokes equation,

The fluid-fluid interfacial stress balance presented in Eq. (2) with viscous, magnetic, and electric stress tensor contributions from Eqs. (3)–(5), respectively, was solved along the fluid interface. The mesh velocity along the interface was subject to the following constraint:

The magnetic field was solved using the magnetostatics module in terms of the magnetic potential *V*_{m}. In a static domain where there is no electric current, the magnetic field is related to the magnetic potential as follows:

From Gauss’s law for magnetism, $\u2207\u22c5B\u2192=0$, the following relation can be derived, which is solved within the simulation domain in conjunction with Eq. (12):

The relative permeability *μ*_{r} is found using the constitutive equation presented in Eq. (8). The following boundary condition is applied to the fluid-air interface to ensure the continuity of the normal component of the magnetic field:

The electric field was solved using the electrostatics module in terms of the electric potential *V*,

subject to

where $D\u2192=\mathit{\epsilon}E\u2192$. The coupling of the electric and magnetic stress tensors with the fluid stress tensor was achieved with the use of a weak form contribution in the following form:

where *test( )* is the test function (determined by COMSOL) and *σ*_{e} and *σ*_{m} are the normal electric and magnetic stress components, respectively. Modeling the fluid as a perfect electrical conductor, the normal electric stress becomes

while the normal component of the magnetic stress is

Note that **T**_{e} and **T**_{m} are the electric and magnetic stress tensors, respectively, and the superscripts + and − denote the fluid above and below the ferrofluid-air interface, respectively. Due to the high conductivity of the ionic liquid carrier fluid, the ferrofluid was modeled as a perfect electrical conductor with an equipotential fluid interface. For a perfect dielectric or perfect conductor, no tangential electric surface stress is present. Boundary conditions on *B*_{n} and *H*_{t} over the fluid interface preclude a tangential magnetic stress. Derivations of Eqs. (18) and (19) as well as the proof of the absence of corresponding tangential stresses are presented in detail in the work of Castellanos^{22} and Rosensweig,^{21} respectively. The Helmholtz field was applied as a boundary condition far from the fluid volume where the contribution to the field from the fluid is negligible. At each solver step, the instantaneous fluid geometry was used to calculate the local magnetic and electric fields.

A mesh validation study was performed with element densities of 350, 700, 1000, and 1500 along the fluid interface for the sharpest peaks investigated. The distribution scheme employed concentrated elements at the apex—growing the elements gradually along the interface towards the base. The refinement study revealed less than a 0.1% variation in the apex height over the meshes investigated, and thus the baseline 350 elements were selected to reduce the computational cost. Adaptive remeshing was integrated into the model such that when the minimum element quality became less than 0.1 (a minimum recommended by the software package), the geometry was remeshed. This enabled large deformations in the geometry. The droplet volume and the element count along the fluid interface were conserved during the remeshing process. Element quality was based on the skew factor of the triangular elements, and the average mesh quality in the domain was always between 0.97 and 0.98.

A dynamic simulation was employed that is in principle capable of providing the temporal response of the fluid. However, our goal was to recover the static solutions, and so the dynamic simulation was used in the following manner. The meniscus was initially defined as a hemisphere, and the magnetic field was set to a constant value. The dynamic simulation then computed the shape relaxation of the hemispherical meniscus to the final (non-hemispheric) static profile governed by the magnetic stress. The simulation ran for 0.2 s, although the droplet profile typically did not evolve any measurable amount after 0.1 s. The interim dynamic response of the meniscus was of little relevance to our study, and thus no information is reported here other than final static configurations. The effect of electric stress was then computed by starting a new simulation that used the previously converged meniscus profile as an initial condition. In this new simulation, the extraction voltage was quickly ramped up to a pre-set value and then held constant for a long-time period compared to the meniscus dynamic response to this changing voltage. Within 0.1 s, the meniscus converged to a new steady-state shape governed by the magnetic and electric stresses that we report as the static solution for this combination of magnetic and electric fields. This process was repeated for a number of different voltages and magnetic fields in order to assemble static results as a collection of multiple dynamic simulations. This process is presented below in Fig. 4.

### C. Simulation results and comparison with data

Both the applied electric field and the applied magnetic field exert stress on the meniscus such that the convex profile is elongated in the direction of the field while the base remains pinned to the capillary exit. The COMSOL model was used to calculate the meniscus shape and from this the “apex height,” which is the dimension from the terminal position of the apex to the capillary exit. Simulation results are shown in Fig. 5, in terms of the electric Bond number $(Be=\mathit{\epsilon}0Ea2R0\u2215\sigma )$ and magnetic Bond number $(Bm=\mu 0H02R0\u2215\sigma )$, where *E*_{a} and *H*_{0} are the apex electric and free space (a Helmholtz coil without the presence of fluid) magnetic fields, respectively, *R*_{0} is the interior radius of the capillary needle, and *Z* is the apex height. For a given voltage, the apex height increases with the strength of the applied magnetic field, which acts in tandem with the electric stress to contribute to the deformation of the meniscus. As the stress is increased, the meniscus transitions into an instability which results in spray emission. The simulation is not capable of predicting the fluid behavior while spraying; however, the run-up to spray onset is evidenced by the asymptotic growth in the apex height. Onset was approximated by the highest voltage at which a stable solution could be obtained. The onset voltage was approached by solving incrementally smaller voltage steps (down to 1 V) while approaching the instability.

Spray onset was easily observed in the laboratory experiment by the sudden appearance of emission current as the bias voltage was slowly increased on the capillary. Three startup tests were performed for each magnetic field. The experimentally measured and simulated onset voltages for the capillary needle are compared in Fig. 6.

In the case of the experimental results, a 200-G magnetic field resulted in a 22% drop in the onset potential. Overall, the model performed well at predicting the onset of emission of the capillary needle emitter for the range of magnetic fields studied. It is instructive to use the model to analyze the relative effects of electric and magnetic stresses. Figure 7 shows the two components of stress for the 200-G magnetic field test case. Near the onset of spray, the electric stress at the meniscus apex is five to ten times larger than the magnetic stress. The primary role of the magnetic stress is to pre-condition the meniscus shape such that threshold to instability occurs at a significantly lower voltage.

## VI. PREDICTIVE MODEL OF ELECTROMAGNETIC SESSILE DROPLET DEFORMATION

The motivation for this research was to understand how electric and magnetic stresses act together to cause spray from the fluid tips formed via the Rosensweig instability. The capillary needle experiment and simulation reported in Sec. V were used to develop and validate a predictive modeling tool. With the foundations of this tool in hand, the second objective was to remove the capillary structure and use the model to predict fluid motion and interface dynamics for an arbitrary droplet and field configuration. We further develop and validate the model through computational and experimental studies of an electrically and magnetically stressed sessile droplet.

### A. Experimental setup

An imaging apparatus was utilized to capture the shape of a sessile ILFF droplet under controllable electric and magnetic fields. This setup, shown in Fig. 8, backlights the ILFF droplet and records silhouette images. This method of imaging allows for precise edge detection.

A Helmholtz coil, shown in Fig. 9, generated a variable uniform magnetic field. The coil consisted of a pair of solenoids, each containing 100 windings. The coil and power supply are capable of generating a maximum field of 310 G. Slots in the coil core allowed for light to pass through to backlight the droplet and provide imaging access. Coil current was measured with a Hall effect current sensor, and a calibration relationship was obtained with the use of an Alpha Labs GM-2 Gauss Meter with a High Stability Universal Probe. The fluid droplet was biased by applying a voltage to the electrode (2) in Fig. 9 using an UltraVolt High Voltage Amplifier (HVA) module with a range of ±5 kV. The biased and grounded electrodes were separated by a distance of 4 mm.

Brass was selected for the electrode material for the desirable wetting characteristics it demonstrated with the ILFF. The fluid volume and contact energy were sufficient to constrain the droplet to a regime where only a single peak would develop. The fluid was placed on the electrode in the presence of a magnetic field. Images were then taken incrementally, increasing the electric field for each subsequent image. A sample image obtained from this system is shown in Fig. 10. Silhouette images were then processed to perform edge detection and volume integration.

Previous work by Gollwitzer suggests the possibility of a magnetic-field-dependent hysteresis in the height of a ferrofluid peak, which was attributed to the fluid wetting the container.^{13} For the ILFF used in this study, hysteresis was observed prior to the droplet becoming pinned, as shown in Fig. 11. It was found that hysteresis could be minimized by reducing the magnetic field to zero following insertion of the droplet. When the field was reapplied, the droplet would be pinned and would still form only a single peak. This pinned peak had negligible hysteresis during subsequent changes in the magnetic field strength.

Operating at peak current, the Helmholtz coil dissipates approximately 150 W of power. Surface tension and magnetization are both strong functions of temperature. To minimize temperature influences, the slide holding the sessile droplet was also thermally insulated from heat sources using Teflon and PLA. To quantify temperature effects, a droplet of the ILFF was imaged under constant fields for 120 s, and the change in height was measured. During the time required to obtain an image set (∼60 s), the height variation was observed to be less than 2% of the initial apex height as can be observed in Fig. 11. This change can be attributed to temperature variations and the low creep velocity of the contact point. In all subsequent testing, data collection was completed within 60 s of applying the magnetic field to render the temperature effect negligible.

### B. Computational approach

The dynamic simulation was designed to solve for temporal meniscus evolution and equilibrium steady-state geometry for arbitrary fields. The computational domain is shown in Fig. 12. The meshing, adaptive remeshing, and simulation approach for this study were identical to the approach utilized in Sec. V B. To reduce the computational load, the physics interfaces were solved only in the necessary domains. The location of the pinning point (denoted by *R* in Fig. 12) was determined by imaging the sessile droplet in the absence of any electric field. The initial fluid geometry within the model was set to have an equivalent volume and contact-plane radius as the imaged droplet. No constrains were placed on the contact angle.

### C. Simulation results and comparison with data

Four sets of laboratory images will initially be discussed in this section. Elements of these sets are presented in Fig. 13. The droplet shown was exposed to four different magnetic field strengths; for each magnetic field, the voltage between the electrodes was increased from 0 to 4000 V in 100 V increments.

Dynamic simulations were performed for each of the four sets of images. The simulated profiles and silhouette images from laboratory data were found to have excellent agreement for voltages up until approximately 85% of the onset voltage, after which the simulations slightly over-predict the meniscus deformation. Full meniscus profile comparisons are shown in Fig. 14, with the results across all simulations and experiments summarized in Fig. 15 by comparing the droplet apex height as a function of the magnetic and electric Bond numbers. *B*_{e} is calculated using the apex electric field, *E*_{a}, which is the field that exists at the tip of the meniscus and is thus strongly dependent upon the meniscus geometry. For the simulated results, retrieval of *E*_{a} is trivial. For the experiment, *E*_{a} was extracted by importing the measured meniscus silhouette geometry into an electrostatic solver as a rigid (non-deformable) conductor and using the electrode geometry and applied potential to calculate the field at the apex. *B*_{m} is calculated using *H*_{0}, which is the vacuum magnetic field created by the Helmholtz coil without the presence of a ferrofluid. The scaling dimension *R*_{0} was set to be the radius of an equivalent-volume hemisphere.

The electric and combined magnetic stress components $(\sigma en=12\mathit{\epsilon}oEn2\u2009and\u2009\sigma mn=12\mu 0Mn2+\mu 0\u222b0HMdH)$ at the meniscus apex for a single simulated droplet under a constant magnetic field are presented in Fig. 16. It can be observed that the combined magnetic stresses dominate at low applied voltages but are rapidly overtaken by the electric stress near the onset of spray. As the magnitude of the applied voltage increases, the electric stress at the meniscus apex grows until it is the dominant perturbation to the interface. This is in part because the apex surface grows closer to the counter electrode, but more strongly because the apex radius of curvature decreases. The magnetic stresses remain relatively constant, compared to the electric stresses, but increase slightly as the applied voltage increases. The individual stress components along the entire fluid interface are presented in Fig. 17 for a single applied voltage.

The fluid magnetic-pressure, $\mu 0\u222bMdH$ term, has a significant role along the entire interface. Meanwhile, the magnetic-normal term, $12\mu 0(M\u2192\u22c5n^)2$, closely resembles the analogous electric-normal stress, $12\mathit{\epsilon}0(E\u2192\u22c5n^)2$, with both terms increasing rapidly and reaching their maximum at the fluid apex.

## VII. CONCLUSION

To conclude, we have dynamically simulated a capillary electrospray emitter under both electric and magnetic stresses. The results of these simulations agree well with the onset measured experimentally for a matching configuration. This study also demonstrated a strong correlation between decreasing onset potential and increasing magnetic field strength. This relation was hypothesized seeing that the magnetic and electric stresses would act in tandem to stretch the fluid meniscus—acting as a positive feedback mechanism which in return enhances both stresses. Using the same simulation technique, we have dynamically simulated a single-peak, normal-field instability for a ferrofluid in the presence of a combined magnetic and electric fields, and this model has been validated by laboratory silhouette images.

## ACKNOWLEDGMENTS

The authors would like to thank the Air Force Office of Scientific Research for their support funding this research. Additional thanks are granted to S. Gowtham, the director of research computing at Michigan Tech, Dr. Jeff Allen of Michigan Tech for his guidance and Dr. Gordon Parker for the use of his COMSOL license. This work was supported in part by a NASA Space Technology Research Fellowship under Award No NNX13AM73H.

The authors declare no competing financial interest.

## NOMENCLATURE

*a*Electrostriction parameter

*B*Magnetic flux density (T)

*D*Electric displacement field (C/m

^{2})*E*Electric field (V/m)

*H*Magnetizing field (A/m)

*M*Magnetization (A/m)

*p*Fluid pressure

**T**_{1,2}Fluid stress tensor (N/m

^{2})**T**_{e}Electric stress tensor (N/m

^{2})**T**_{m}Magnetic stress tensor (N/m

^{2})**T**_{s}Stokes stress tensor (N/m

^{2})*V*_{c}Critical voltage for onset of emission (V)

*β*Langevin fit parameter (A/m)

*ε*_{0}Permittivity of free space (F/m)

*μ*Viscosity (mPa s)

*μ*_{r}Relative permeability

*ρ*Fluid density (kg/m

^{3})*σ*Surface tension (N/m)

- $\sigma en$
Surface normal electric stress (N/m

^{2})- $\sigma mn$
Surface normal magnetic stress (N/m

^{2})- $\u25bf$
_{t} Gradient along surface tangent