In this study, we conducted experiments to explore the potential of a low-power exploding foil initiator for accelerating microparticles through high-speed membrane deformation. This involved the use of a conductive layer with a conversion section known as a “bridge,” which was positioned between the substrate and the cover layer. The application of pulsed electrical energy led to Joule heating at the bridge, while the vaporized gas generated impulsive loading, resulting in the deformation of the cover layer. According to the principles of energy conservation, 8.7% of the electrical input energy was converted into kinetic energy for the membrane. This deformation process achieved a velocity of 800 m/s, with a corresponding strain rate of 1.6 × 107 s−1. The applied impulse predominantly induced extension stresses in the cover layer rather than bending stresses. Under these conditions, a 17.5-µm radius polylactic acid bead was propelled and subsequently captured by a silicone gel layer, resembling human dermic skin. Considering factors such as particle clustering and deceleration due to air resistance during supersonic flight, assuming a normal incident angle, it was estimated that approximately half of the ejected particles could reach the human dermic layer, located 200 µm beneath the skin surface. These findings suggest that pulse discharge is a promising method for inducing high-speed membrane deformation, and the electrical microparticle accelerator holds potential for applications in needle-free drug delivery.

Small-object acceleration is a standard and critical technology in several science research fields, such as crater formation in planetary events,1 needle-free drugs,2 and gene3 delivery in medical/agricultural applications. In engineering, surface-reforming applications accelerate particles with diameters of several micrometers (microparticles) using high-speed gas jets. Accelerated microparticles collide with the target and are plastically deformed by their kinetic energy, as observed using the cold-spray method.4 When the kinetic energy is lower than a threshold value, the target material is eroded instead of undergoing particle bonding, as observed in sandblast treatment.5 An exploding foil initiator (EFI) also accelerates a small object, which is referred to as a “flyer,” at several kilometers per second to act as a shock switch.6 To study material phase changes under pulsed dynamic pressure, the accelerated flyer collides with an explosive for initiation or creates a high-pressure field in the range of 109 Pa. For more fundamental aspects, Lee et al. developed laser-induced micro-projectile impact testing (LIPIT), which accelerated a single microparticle in the range of 103 m/s using an expanding laser ablation gas7 to understand the dynamic mechanical behavior of materials subjected to high-velocity impact.

Although the microparticle acceleration technique is a powerful and attractive technology in various research fields, particle acceleration schemes can only be divided into two types, as reviewed in Ref. 8. The first scheme utilizes an expanding high-pressure gas that injects particles into a flowing gas or expanding vaporized metal. The kinetic energy of the ejected particles is directly dependent on the gas velocity, charged gas pressure, input electrical energy, and irradiating laser energy, which are the control parameters in cold spray/sandblasting, EFI, and LIPIT without an elastomer layer,7 respectively. The particle velocity increases with increasing input energy, and there are no physical limitations. The second acceleration scheme utilizes high-velocity membrane deformation. Similar to the first scheme, the driving force of membrane deformation is a high-pressure gas created by a shock tube, explosive media, or short-pulse laser irradiation. However, high-pressure gas is confined between the substrate and membrane instead of undergoing free expansion. Due to the pressure confinement effect,9 the membrane experiences high-strain deformation in the range of more than 106 s−1,10 and the particles deposited onto the membrane surface have an initial ejection velocity that corresponds to the membrane deformation velocity. In contrast to the first scheme, the ejected particle(s) are not exposed to high temperatures or even ionized driver gases. Therefore, the second scheme can maintain the original temperature and/or suppress the deactivation of the chemical properties. In addition, the membrane acts as a shield to protect the target from flowing/expanding gas or fragments produced during device operation. However, the deformation velocity is physically limited by the membrane material properties of impulsive loading to prevent fragmentation. Recently, Kim and Jang reported the minimum threshold energy of the EFI operation required to create a flyer under the given dimensions of the device.11 Their numerical results showed that the membrane was deformed by more than 500 m/s when the input power was lower than the threshold energy required for flyer creation. For the intended purpose of EFI, fragmentation should not be avoided. However, considering that the typical particle velocity driven by a shockwave from the shock tube or blast wave from the explosive media is in the subsonic range8 (≤340 m/s), low-power (lower than the threshold energy for fragmentation) EFI operation is not a failure condition but a desired condition for particle acceleration by a membrane deformation.

This study investigated the potential use of low-power EFI operation as an electrical microparticle accelerator. The configuration of the investigated device was identical to that of a conventional EFI device. As the electrical input power was changed, we measured the membrane deformation velocity and estimated the breakdown of the applied impulse. We also demonstrated a microparticle ejection to a “dermic” silicone gel target for possible usage of needle-free drug delivery application.

Figure 1(a) shows a schematic of the particle accelerator assembly. The accelerator contained three layers: (1) a substrate layer (FR-4), (2) a conductive layer (copper), and (3) a cover layer (polyimide). As shown in Fig. 1(b), the conductive layer and the cover layer included a converged square-shaped section (“bridge”). The dimensions and masses of the conductive and covered bridges are listed in Table I. Both bridges had the same lengths l but different thicknesses h: the conductive bridge had a single thickness of 9 µm, whereas that of the cove bridge was valid for 25 and 50 µm. The designed mass of the bridge was calculated as m = ρl2h.

FIG. 1.

Schematics of experimental apparatus in this study: (a) particle accelerator, (b) photograph of the accelerator with an enlarged view of the bridge section, and (c) electrical circuit with diagnostic equipment for current, voltage, and deformation velocity.

FIG. 1.

Schematics of experimental apparatus in this study: (a) particle accelerator, (b) photograph of the accelerator with an enlarged view of the bridge section, and (c) electrical circuit with diagnostic equipment for current, voltage, and deformation velocity.

Close modal
TABLE I.

Designed dimensions and masses of the conductive layer and the cover layer.

ItemParameterSymbolUnitValue
Conductive layer Material ⋯ ⋯ Copper 
 Density ρb kg/m3 8.9 × 103 
 Length lb mm 0.3, 0.4 
 Thickness hb μ9.0 
 Mass mb kg 7.2 × 10−9, 1.3 × 10−8 
Cover layer Material ⋯ ⋯ Polyimide 
 Density ρc kg/m3 1.4 × 103 
 Length lc mm 0.3, 0.4 
 Thickness hc μm 25, 50 
 Mass mc kg 3.2 × 10−9–1.1 × 10−8 
ItemParameterSymbolUnitValue
Conductive layer Material ⋯ ⋯ Copper 
 Density ρb kg/m3 8.9 × 103 
 Length lb mm 0.3, 0.4 
 Thickness hb μ9.0 
 Mass mb kg 7.2 × 10−9, 1.3 × 10−8 
Cover layer Material ⋯ ⋯ Polyimide 
 Density ρc kg/m3 1.4 × 103 
 Length lc mm 0.3, 0.4 
 Thickness hc μm 25, 50 
 Mass mc kg 3.2 × 10−9–1.1 × 10−8 

Figure 1(c) shows the experimental setup for measuring the cover layer deformation velocity measurement with its electrical circuit. The accelerator was connected to the capacitor with flexible printed circuits in parallel [see also Fig. 1(b)]. A high-voltage direct-current power supply was used to charge the capacitor. The charged energy (Ec) ranged as 23–84 mJ. After the power supply charged the condenser, the switch was turned on to connect the devices serially for the initiation. The discharge current J and voltage V were measured using a voltage probe (THDP0100, Tektronix, Inc.) and Rogowski coil (CWT UM/30 B/1/80, Power Electronic Measurements Ltd). The cover layer deformation velocity was measured using a commercial modular photonic Doppler velocimetry (PDV12) system (VH17, IDIL Fibres Optiques) with a bandpass filter (TBF-1550-1.0-FCAPC, Newport Corporation) having a central wavelength and bandwidth of 1550 and 0.8 nm, respectively. For adjusting the probe center with respect to the bridge center, a PDV probe with an outer diameter and working distance of 3.2 and 20 mm, respectively, was set in the PDV probe holder with two gauge pins. The distance between the probe head and bridge surfaces was 8 mm, which was within the working distance of the probe. To record the discharge current, discharge voltage, and PDV signal, two oscilloscopes (1 and 2) were used. Oscilloscope 1 (MDO4024C, Tektronix, Inc.), with a sampling rate of 1.25 GHz, recorded the discharge current and voltage of the condenser. Oscilloscope 2 (MSO64B, Tektronix, Inc.), with a sampling rate of 50 GHz, recorded the discharge current and the PDV probe signal. In both the recordings, the discharge current was used as the trigger signal. All the experiments were conducted at least twice, and reproducibility was confirmed.

The typical time histories of the consumed power P (=JV) in the bridge, cover layer deformation velocity u, and net force fnet applied to the cover layer are shown in Figs. 2(a)2(c), respectively. The time origin (t = 0) corresponds to the time at which the switch is turned on. The operation conditions of Fig. 2 are summarized in the figure caption. Due to the parasitic inductance of the electrical circuit, the P value oscillated between 3.0 and −1.3 MW. Here, a negative P value indicates power reflection from the bridge due to the counter-electromotive force. In contrast to the conventional transient phenomenon of condenser discharge, a simple attenuation wave was not observed because the resistance at the bridge section continuously changed as the bridge temperature was changed by Joule heating. Through oscillation, the condenser-charged energy Ec was gradually consumed at the bridge for 0.63 µs.

FIG. 2.

Typical time history of (a) consumed power at bridge P, (b) surface velocity of the cover layer u, and (c) applied net force fnet (=ρchcdu/dt) on the cover layer: l = 0.4 mm, hc = 25 µm, and Ec = 43 mJ. The fitting function for u and the calculated fnet are depicted in (d) and (e), respectively.

FIG. 2.

Typical time history of (a) consumed power at bridge P, (b) surface velocity of the cover layer u, and (c) applied net force fnet (=ρchcdu/dt) on the cover layer: l = 0.4 mm, hc = 25 µm, and Ec = 43 mJ. The fitting function for u and the calculated fnet are depicted in (d) and (e), respectively.

Close modal

After releasing condenser energy, the cover layer began to undergo plastic deformation with surface velocity u. In Fig. 2(b), the gray circle represents the experimentally measured u value. Each symbol has an error bar from the short-term Fourier transform (STFT) of the PDV signal analysis.13 The time width of the STFT, which corresponds to the effective time resolution u, was set to 10 ns. Under the conditions presented in Fig. 2, the uncertainty in velocity was in the range of 10–15 m/s. As shown in Fig. 2(b), at first, u slightly increased for t ≤ 0.20 µs, and in 0.20 < t ≤ 0.63 µs, u exhibited rapid increment to reach the maximum velocity umax at t = 0.63 µs, which is identical to the time duration of condenser discharge. For t > 0.63 µs, u began to decrease, and at t = 1.5 µs, the cover layer deformation was completed. To estimate the umax value, we define the fitting function as the summation of the four Gaussian functions. The fitted curve is shown in Fig. 2(d). Under these experimental conditions, umax was estimated to be 289 m/s. The strain rate umax/hc was calculated as 1.1 × 107 s−1, which is the same order as the laser-induced membrane deformation (106–108 s−1).10 

During the deformation process, the cover layer experiences both acceleration and deceleration pressures, which arise from the expanding pressure generated by the vaporized conductive layer and internal stresses, including bending and extension phenomena. To calculate the net force per unit area (fnet), we applied the momentum conservation law and derived it from the time deviation of u, given as fnet = ρchcdu/dt. Figures 2(c) and 2(e) display the calculated fnet using a central difference scheme based on experimental data and analytical deviation, respectively. Two distinct positive pressure peaks were observed: fnet = 4.4 N/mm2 at t = 9.5 × 10−2 µs and 39 N/mm2 at t = 0.47 µs. The net impulse density (inet), which represents the time integration of the positive fnet values, was determined to be 9.4 Pa s. The second positive pressure was the primary force responsible for extending the cover layer. The duration of this second positive pressure was estimated to be 0.43 µs, which accounts for less than one-third of the total deformation time, indicating that electrical membrane deformation can be treated as an impulsive phenomenon.14 

To understand the relationship between P, u, and fnet, we established two time zones: Time-zone 1 (0 ≤ t ≤ 0.12 µs) and Time-zone 2 (0.20 µs ≤ t ≤ 0.63 µs) corresponding to the positive input power ranges marked by gray hatching in Fig. 2. In Time-zone 1, the input energy E1, calculated by time integration of P, amounted to 22 mJ. Considering the bridge's mass with lb = 0.4 mm and hb = 9 µm, E1 exceeded the heat of fusion for the bridge material by more than eightfold. The melted bridge material exerted pressure on the cover layer, resulting in fnet = 4.4 N/mm2, and u increased to 16 m/s. Between Time-zones 1 and 2, fnet decreased to zero due to the reflection of condenser discharge power, which was not absorbed by the bridge section. In Time-zone 2, the discharge power was once again absorbed by the bridge section. For the calculation of E1, the input energy in Time-zone 2, E2, amounted to 21 mJ. The total input energy, E1 + E2, surpassed the combined values of the heat of fusion and evaporation. The vaporized bridge material rapidly expanded, propelling the cover layer as the applied fnet reached 39 N/mm2, ultimately achieving maximum velocity.

Figure 3 shows the time history of deformation velocity u with different specific energies e (≡Ec/mb) with l = 0.4 mm and hc = 25 µm. The symbols and solid lines represent experimental data and curve fitting, respectively. Regardless of e values, the time duration for cover layer deformation was identical for 1.3 µs. The umax was strongly dependent on e such that when e increased 1.3 times (from 7.8 to 10.5 MJ/kg), umax increased by 8.0 times (from 88 to 702 m/s). The maximum strain rate was 2.8 × 107 s−1 for e = 10.5 MJ/kg. In addition, we confirmed that, when e was lower than 7.8 MJ/kg, the PDV diagnostics detected no signal because the deformation velocity was excessively slow. However, when the e value was higher than 10.5 MJ/kg, the cover layer ruptured, and the fragment flowed at more than 1.0 km/s, which is the desirable operating condition for the exploding foil initiator.

FIG. 3.

Time history of deformation velocity u with different specific energies e: lc = 0.4 mm and hc = 25 µm. Symbols and solid lines represent experimental data and curve fitting, respectively.

FIG. 3.

Time history of deformation velocity u with different specific energies e: lc = 0.4 mm and hc = 25 µm. Symbols and solid lines represent experimental data and curve fitting, respectively.

Close modal
The e dependence of umax for different cover bridge dimensions is shown in Fig. 4. Regardless of the cover bridge size lc, the measured umax values were divided into two cases with cover layer thicknesses hc (red and blue symbols in Fig. 4). In each case, umax increased with e. For hc = 25 and 50 µm cases, the umax reached 702 and 800 m/s, respectively. The maximum strain rate was 2.8 × 107 s−1. The obtained umax was more than twice that of deformation techniques using shock waves15 (∼335 m/s), blast waves from micro-explosions16 (∼166 m/s), and pulsed laser irradiation17 (∼292 m/s). Therefore, we concluded that electrical pulsed energy is an attractive driving force for high-speed membrane deformation. Based on the specific energy conservation law, the obtained umax values were fitted using the following equations:
mbe=12mcumax2+0hc12ρblb2ub2dh+mbe0+mces.
(1)
The first and second terms on the right side represent the kinetic energies of the cover and conductive bridges, respectively. The velocity of the conductive bridge is denoted as ub. The third and last terms represent the energy losses. where e0 is the specific energy required to start the cover layer deformation, including the latent heat of copper and the deformation resistance of polyimide. Therefore, e0 is dependent on hc. Furthermore, es represents other energy losses through deformation by strain or frictional forces. Assuming a linear velocity distribution inside the conductive bridge,18  ub = umax h/hc, the following equation is obtained:
umax=2ηee0mc/mb+1/3.
(2)
The coefficient η is defined as
η12mcmb+13umax2/12mcmb+13umax2+mcmbes.
(3)
Using Eqs. (2) and (3), the obtained umax values were fitted via η and e0 for different hc values, as shown in the broken lines of Fig. 4. The mass ratios and fitting results are summarized in Table II with different hc values. We newly introduced a parameter η′ defined as
ηηmc/mb+1/3.
(4)
Substituting into Eq. (2), we obtain
umax=2ηee0.
(5)
Equations (4) and (5) indicate that the parameter η′ is an energy conversion efficiency from the energy stored in an object with mass mb, namely, mb(ee0), to the kinetic energy of an object with mass mc + mb/3, namely, ½(mc + mb/3)umax2. The calculated η′ are also shown in Table II. Regardless of hc values, η′ was identical to 8.5%.
FIG. 4.

Maximum deformation velocity umax with different bridge dimensions. Broken lines represent curve fitting by Eqs. (4) and (5).

FIG. 4.

Maximum deformation velocity umax with different bridge dimensions. Broken lines represent curve fitting by Eqs. (4) and (5).

Close modal
TABLE II.

Mass ratio and fitting results of Eq. (2) with different hc values.

hc (μm)mc/mbe0 (MJ/kg)η (%)η′ (%)
25 0.44 7.7 6.6 8.5 
50 0.87 10.3 10.3 8.5 
hc (μm)mc/mbe0 (MJ/kg)η (%)η′ (%)
25 0.44 7.7 6.6 8.5 
50 0.87 10.3 10.3 8.5 
The cover layer deformation caused by vaporized copper gas was treated as a biaxial state of stress applied to a thin plate. The extension and bending behaviors can be analyzed for both normal and shearing stresses from the energy solution.19 By equating the strain and kinetic energies, we obtain the following equations:
ineti02=σ̄b,n+σ̄b,s+σ̄e,n+σ̄e,s=π41+XY2δchc+23XYδchc+3π2641+XY2δchc2+43XYδchc2
(6)
The four dimensionless terms on the right side of Eq. (6) are associated with normal bending stresses σ̄b,n, bending shear stresses σ̄b,s, normal extensional stresses σ̄e,n, and extensional shear stresses σ̄e,s, respectively. The characteristic impulse density i0 on the left-hand side is defined as
i0ρcσhc2/X.
(7)
Here, we assume a uniform impulse such that the impulse distribution within the bridge is ignored. The values X and Y are the half length of the cover bridge, X = Y = lc/2, and σ′ is a tensile strength (=69 MPa for polyimide). As described in Sec. III A, the net impulse density inet was calculated using the time integral of the positive pressure [Fig. 2(c)]. The displacement δc was calculated by the time integration of u during the deformation [see Fig. 2(b)]. Figure 5 shows the measured (symbols) and predicted (broken lines) deformations in a uniformly impulsed cover bridge with different dimensions. Regardless of the lc and hc values, the calculated δc was consistent with the prediction curve obtained using Eqs. (6) and (7), respectively. Therefore, cover layer deformation can be understood using the theory of a uniformly impulsed rectangular plate.
FIG. 5.

Normalized displacement δc/hc in a uniformly impulsed cover bridge with different dimensions. The broken line is calculated from Eqs. (6) and (7).

FIG. 5.

Normalized displacement δc/hc in a uniformly impulsed cover bridge with different dimensions. The broken line is calculated from Eqs. (6) and (7).

Close modal

The breakdown of the bending and extending stresses, depicted on the right-hand side of Eq. (6), is shown in Fig. 6 as functions of the normalized specific energy e/e0. Here, σ̄t is calculated as σ̄t=σ̄b,n+σ̄b,s+σ̄e,n+σ̄e,s. Regardless of the e/e0 value, the extensional shear stresses σ̄e,s was the main component of σ̄t. When the specific energy was close to e0 (e/e0 = 1.0), stresses, except for σ̄e,s, contributed ∼20% for σ̄t. As e/e0 increased, σ̄b,n and σ̄b,s decreased, whereas σ̄e,n increased. At e/e0 = 1.46, the relative rate of both σ̄b,n and σ̄b,s was less than 5% and that of σ̄e,n and σ̄e,s was 26% and 64%, respectively. Because the sum of the extensional stresses (σ̄e,n+ σ̄e,s) was 90%, the applied impulsive force onto the cover bridge was consumed to induce extensional stresses.

FIG. 6.

Breakdown of the dimensionless bending and extending stresses σ̄ as a function of normalized specific energy e/e0.

FIG. 6.

Breakdown of the dimensionless bending and extending stresses σ̄ as a function of normalized specific energy e/e0.

Close modal

In this section, we demonstrate microparticle acceleration by high-speed membrane deformation for medical applications such as drug delivery to human skin.2 One possible application is a suspension of injections for mammary and prostatic cancers. Commonly, a suspension injection uses biodegradable polymer-made microcapsules (diameter of several tens of micrometers), wherein the chemical components are enclosed. We employed a polylactic acid bead (50-00-304, Corefront Corp., mass density ρp = 1.2 × 103 kg/m3) as a test particle. The particles had a mean radius of 17.5 µm and maximum and minimum radii were controlled for 25 and 10 µm, respectively. The beads had a fluorescent mark to distinguish fragments of the cover bridge. The maximum excitation and emission wavelengths were 552 and 580 nm, respectively. A schematic of the particle acceleration/ejection is shown in Fig. 7. The acceleration procedure involves three steps. First, microparticles were deposited onto a cover bridge (Step 1). Subsequently, the switch was turned on to discharge the stored energy of the condenser, and the cover bridge deformed with velocity u, as described in Sec. III A (Step 2). In addition, the deposited particles moved at velocity u. When the deformation velocity reached the maximum value u = umax, the deposited particles were dislodged.17 This is because the resistance and friction forces only affect the cover bridge and the inertial force dominates the particle motion. Subsequently, the ejected particles underwent ballistic flight and finally penetrated the target (Step 3).

FIG. 7.

Schematic of particle acceleration and ejection procedure by the high-speed membrane deformation.

FIG. 7.

Schematic of particle acceleration and ejection procedure by the high-speed membrane deformation.

Close modal

Figure 8 shows an image of the deposited particles obtained using a digital microscope (DN-117M-JP, Llutico). A micropipette (00-NPP-10, Nichiryo Co., Ltd.) was used to suck 1.0 µl of the particle suspension (10 mg/ml), which was discharged onto the cover bridge with lc = 0.4 mm and hc = 50 µm. After 15 min of air drying, the particles were distributed in a 1.35 × 0.77 mm2 area, as shown in the yellow box. Among the deposited particles, only those lying on the cover bridge accelerated. Considering the ratio between the particle deposition area estimated by Image J Fuji and the cover bridge area (=lc2), the total mass of the ejectable particles was ∼1.0 × 10−9 kg.

FIG. 8.

Microscopic image of the deposited particles distributed within a 1.35 × 0.77 mm2 area as depicted in the yellow box.

FIG. 8.

Microscopic image of the deposited particles distributed within a 1.35 × 0.77 mm2 area as depicted in the yellow box.

Close modal
The ejected particles were captured using a 6-mm diameter human “dermic” silicone gel (TANAC Co. Ltd., mass density ρt = 9.5 × 102 kg/m3) that was set 2 mm away from the cover bridge. The alignment of the cover bridge and silicone gel was adjusted using gauge pins, as shown in Fig. 1(c). The specific energy e was set to 15 MJ/kg and umax = 800 m/s. With this velocity, the kinetic energy of each particle Ψp was 5.6 × 10−6 J. Rakesh et al.16 estimated the energy loss through the particle ejection procedures. When the particles are dislodged from the cover bridge surface, a small amount of energy is lost in overcoming van der Waal’s energy of adhesion Ψvw between particles/surface. The Ψvw can be estimated as20 
ψvw=H6rpξ+rp2rp+ξ+lnξ2rp+ξ,
(8)
where H, rp, and ξ are Hamaker’s constant (=2.0 × 10−14 J), particle radius (=17.5 µm), and distance between the particles and surface (=2.6 × 10−11 m), respectively. By substituting these values into Eq. (8), Ψvw is calculated as 1.9 × 10−9 J for each particle, which is 0.03% of Ψp. Therefore, the energy loss during the particle dislodging procedure was negligible, and the initial particle velocity up,i, was identical to umax.

Figure 9 shows a microscopic overlay image of the particles captured using a fluorescence microscope (BZ-X800, Keyence Corp.). Because both the bright-field and fluorescence images showed excellent agreement, the detected objects were microparticles only and there were no fragments of the cover bridge. Figure 9 also shows histograms of the relative positions of each particle along the horizontal and vertical directions, as estimated from image analysis using ImageJ Fuji. The histograms for each axis display data exclusively for that axis and do not incorporate information from the other axis. The ejected particles exhibited a distribution spanning ±1.4 mm along the horizontal axis and ±1.6 mm along the vertical axis. Consequently, the deposited particles were propelled from the cover bridge in all directions, consistent with Ref. 15. The distribution of captured particles depends on two key factors: (i) the initial distribution of deposited particles onto the cover layer and (ii) random collisions during their ballistic flight toward the target. During the drying of particle suspensions, capillary flow plays a role in transporting particles from the droplet’s center to its periphery, resulting in a ring-like deposit.21 The extent of deviation in the positions of deposited particles is influenced by concentration, with highly concentrated suspensions inhibiting capillary flow, thereby promoting uniform distribution as the solution dries. Shimobayashi et al. have suggested that adding sugar to the particle suspension can achieve a uniform distribution of particles on the cover layer.22 On the other hand, random collisions among particles during their flight contribute to a uniform distribution of captured particles. However, determining the precise influence of these two effects is a challenging task. For applications involving suspension injection, the primary concern is typically the penetration depth, and the particle distribution in a horizontal plane would not be an essential matter.

FIG. 9.

Microscopic overlay image of the “dermic” silicone gel after particle ejection. The histograms of each particle’s relative positions along the horizontal and vertical directions are also shown.

FIG. 9.

Microscopic overlay image of the “dermic” silicone gel after particle ejection. The histograms of each particle’s relative positions along the horizontal and vertical directions are also shown.

Close modal

The image analysis also introduced a projected area of each particle Ap, and we defined the equivalent radius rp,equ as rp,equ ≡ (Ap/π)½. The normalized histogram of the estimated rp,equ is shown in Fig. 10 (gray box), with curve fitting using a Gaussian function (red line). As mentioned above, the particle originally had a mean radius of 17.5 µm (the black arrow in Fig. 10), and the maximum and minimum radii were controlled for 25 and 10 µm, respectively. Because the deposited particles formed clusters, the obtained histogram had a mean value at rp,equ = 22.5 µm and larger rp,equ than the maximum rp observed. However, a smaller rp,equ suggests that the particle collapses during ballistic flight, as observed in Ref. 23.

FIG. 10.

Normalized equivalent particle radius rp,equ distribution (gray bar) with Gaussian curve fitting (red line) on the left axis. The black arrow indicates the mean radius before ejection. Additionally, the calculated penetration depth zp from Eq. (9) is depicted (blue broken line) on the right axis.

FIG. 10.

Normalized equivalent particle radius rp,equ distribution (gray bar) with Gaussian curve fitting (red line) on the left axis. The black arrow indicates the mean radius before ejection. Additionally, the calculated penetration depth zp from Eq. (9) is depicted (blue broken line) on the right axis.

Close modal
The penetration depth zp is estimated from the Poncelet equation,24,
zp=4ρprp,equ3ρtCD,tlogρtCD, tup,fcosθ22R+1,
(9)
where up,f is the impact velocity, θ is an incident angle, CD,t is the drag coefficient of the target, and R is the yield resistance. In this estimation, we set CD,t = 1.5 and R = 21 MPa, as reported in Ref. 25. As the first approximation, we assumed a normal incident angle (cos θ = 1.0) in the following discussion. To estimate up,f, the drag force from the ambient air affected by the ballistic flight should be considered. The ejected particle experiences a viscous resistance force, FD, and the equation of motion is as follows:
mpdupdτ=FD=12CD,aρaup2πrp,equ2,
(10)
where up is the particle velocity, τ is the time after the dislodging, and ρa is the mass density of air. Carlson and Hoglund26 developed the drag coefficient of air CD,a as a function of the Mach number M (≡u/ua) and Reynolds number Re (≡ρaurp/μa) as follows:
CD,a=24Re1+0.15Re0.6871+exp(0.427/M4.633/Re0.88)1+M/Re3.82+1.28exp1.25Re/M,
(11)
where μa and ua are the viscosity and speed of sound in air, respectively. For a temperature of 20 °C and an atmospheric pressure of 1.0 × 105 kPa, ρa = 1.3 kg/m3, μa = 1.8 × 10−5 Pas, and ua = 341 m/s. The particle position xp was calculated from the time integration of up using the fourth-order Runge–Kutta method. The impact velocity, up,f was the velocity at xp = 2.0 mm. Typically, up,f = 769 m/s for mean rp,equ. Figure 10 shows the estimated zp for each rp,equ value (blue broken line) on the right axis. The calculated zp exhibited a linear dependence on rp,equ because zp depends linearly on rp,equ, whereas the nonlinear dependence on rp,equ through CD,a and up,f exhibited a limited effect. From the perspective of suspending injection usage, the penetration depth should be deeper than the dermic layer, which is ∼200 µm under the skin surface.27 As shown in Fig. 10, zp exceeded this threshold at rp,equ = 22.5 µm (mean value of Gaussian distribution), and we can expect that half of the ejected particles would reach the dermic layer. In this estimation, we ignored the effect of the incident angle, and the penetration depth would be overestimated. To validate zp estimation, precise measurements using a confocal fluorescence microscope, as conducted in Ref. 28, are desired, which is left for future work.

In this study, we investigate the potential of a low-power exploding foil initiator as an electrical microparticle accelerator. The accelerator contains a substrate, a copper conductive layer, and a polyimide cover layer. The conductive land cover layers included a converged square-shaped section (“bridge”). When the condenser-charged energy was released, intensive Joule heating occurred at the bridge, and the conductive bridge was vaporized. Consequently, the cover layer experienced a high-speed deformation driven by the confined copper gas. Under the threshold level for fragmentation, the deformation velocity increased with increasing input energy up to 800 m/s, which is more than two times faster than that reported in previous studies. Therefore, pulsed discharge is an attractive method for driving high-speed membrane deformation. The energy transfer from the input electrical energy to the membrane kinetic energy was 8.5%, irrespective of the bridge dimensions. Based on the biaxial state of stress, 90% of the stresses induced by the applied impulsive loading were contained as surface extensions, and the other 10% were surface bending. In addition, we demonstrated needle-free drug delivery applications as a possible usage of the proposed electrical microparticle accelerator. Because the theoretically estimated energy loss through the particle dislodging process is negligible, the particles deposited onto the cover layer obtained an initial velocity corresponding to the maximum cover layer deformation velocity, as previously mentioned. Considering particle clustering and deceleration by the drug force during supersonic ballistic flight with an assumption of normal incident angle, half of the ejected particles would reach the human dermic layer that is 200 µm below the skin surface. These results indicate that the electrical microparticle accelerator can facilitate needle-free drug delivery. Although existing research supports our assumptions and estimations, precise penetration-depth measurement using a confocal fluorescence microscope is desired, which is left for future work.

This study was supported by the JST FOREST Program (Grant No. JPMJFR2058, Japan). Kentaro Hayashi and Hidenari Naito (Ricoh Elemex Corp.) were technically supported, particularly in electrical circuits. The authors acknowledged enlightening discussion with Akihiro Sasoh (Nagoya University).

The authors have no conflicts to disclose.

D. Ichihara: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Funding acquisition (lead); Investigation (lead); Methodology (lead); Project administration (lead); Resources (lead); Software (lead); Supervision (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). R. Takata: Data curation (equal); Methodology (equal).

The data that support the findings of this study are available within the article.

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