Binder droplet impact mechanism on a hydroxyapatite microsphere surface in 3D printing of bone scaffolds

Abstract

The combination of hydroxyapatite composite powder and three-dimensional (3D) printing rapid prototyping techniques has markedly improved skeletal interactions in orthopedic surgery applications. 3D printing methodology ensures effective bionic microstructure and shape interactions between an implant and the surrounding normal tissue. In effort to enhance the quality, precision, and mechanical properties of printed bone scaffolds, this study examines binder droplet spreading performance on the surface of hydroxyapatite (HA) microspheres. The piezoelectric nozzle diameter is about 10 μm, which sprays droplets 20 μm in diameter. The average size of HA powder particles is about 60 μm in diameter. Most laboratories, however, are limited to observation of a single droplet 20 μm or smaller in diameter impacting a spherical surface 60 μm in diameter. Based on non-dimensional scale similarity theory in axisymmetric Stokes flow dynamics, this study conducted experiments and simulation on the same collision conditions (droplet 200 μm in diameter, spherical surface 600 μm in diameter). Simulation results were consistent with experiment data, and form a basis for future research on modeling droplet impact on spherical surfaces.

Introduction

Researchers, developers, and medical professionals involved in bone tissue engineering have shown increased demand for artificial bone scaffolds; more specifically, for bone substitute, that is as sustainable as possible without sacrificing high quality mechanical, chemical, and biocompatibility properties. Three-dimensional printing (3DP) is a promising method that employs inkjet printing technology to bind bioceramic powder materials to fabricate human-specific, individual substitutes. Hydroxyapatite (HA) [1] is a typical bioactive material which occupies 75 wt% of human bone, regardless of water content. It is highly biocompatible, non-toxic, and able to promote bone growth and conjoin successfully with human bone. Both Mooney [2] and Felicity [3] proved hydroxyapatite [HA, Ca₁₀(PO₄)₆(OH)₂] to be a highly effective synthetic bone replacement material for artificial bone scaffolds, due to the material’s favorable biocompatibility and osteoconductive properties. Previous studies [4, 5] have shown that the smaller the HA powder size, the more biocompatible the HA-implanted prosthesis. Suitable HA powder particle size is a key component of the biocompatibility and mechanical properties of an artificial bone scaffold. Lorna et al. [6] asserted that the porosity and mechanical properties of a bone scaffold are the most important factors in building bionic artificial bone. Bone scaffolds provide such functions as temporary mechanical support, osteoconductivity, and tissue regeneration.

In the bone scaffold fabrication experiments conducted in this study, average powder size was about 60 μm in diameter. Spraying biobinder volume on the surface of the HA powder bed is a crucial point in the fabrication process that controls the mechanical and porosity properties of the scaffold; for typical experimental assessment instruments, however, it is practically impossible to capture an image of a microdroplet flying a distance of 1 mm with comparably high velocity (above 1.0 m s⁻¹). Based on similarity theory, this study built a droplet spray system to observe a single biobinder droplet of 200 μm or larger in diameter impacting a spherical surface of HA microsphere 600 μm in diameter. Comparison between simulation and experimental results presented here can improve inkjet spraying processes and bone scaffold fabrication processes in future by outlining methods for controlling biobinder droplet pattern.

The deposition of a liquid droplet impacting a solid surface is a topic of considerable importance in a variety of applications in aerospace, material science, manufacturing, and chemical industries, as it is related to inkjet printing, spray cooling and deposition, and pesticide spraying. To this effect, a substantial body of theoretical, experimental, and simulation studies on the topic have been conducted in recent years. Roux and Cooper-White et al. [7], for example, studied the spreading of a water droplet impacting a glass surface at different impact velocities, with particular attention to the dynamics of contact angle and its relationship to maximum expanding radius and capillary number. Their results clearly showed that conservation of energy alone is insufficient to describe the dynamics of impact events. Luo et al. [8] came to conclusion that droplet diameter decreased, but the ratio of droplet diameter to nozzle diameter increased rapidly, when nozzle diameter decreased to 100 μs. Pan et al. [9] investigated collisions between water droplets with dimensionless Weber number up to 12,000 and a solid plate with varied surface roughness. Fujimoto et al. [10] experimentally and numerically studied the collision of a liquid droplet with a hemispherical droplet on a horizontal solid surface using the volume of fluid (VOF) method, and the effects of the incoming droplet’s impinging angle on the deformation process. Cao et al. [11] observed molten droplets generated through a graphite nozzle with a 0.3 mm diameter orifice by argon pulse pressure and found that droplets were typically generated at a rate of 1–5 droplets per second. Yarin et al. [12] provided a comprehensive review of previous theoretical and experimental studies on drop impact dynamics, and came to the conclusion that the collision of a liquid droplet on a solid surface is determined primarily by inertia, viscous forces, surface tension forces, and surface properties of the solid surface.

Most previous studies [13–15] focus on the impact of a droplet on a flat surface, and relatively few studies have investigated the impact on a spherical surface. Kim et al. [16, 17] investigated the mechanism of droplet formation from piezoelectric-type inkjet heads by simulating the two-phase flows of air and metal inks. They found that a buffer region helps to maintain stable meniscus features after ejection. Modifying the displacement waveform from the piezoelectric actuator makes it possible to control droplet size, ejection speed, and satellite droplets. Bakshi et al. [18] studied the effects of droplet Reynolds number and target-to-drop size ratio on the impact of a drop on a spherical surface. Li and Wang [19] developed a numerical model to simulate the deposition of a droplet with low-impact energy impinging onto a spherical surface, with special emphasis on the effects of impact velocity and impacted-sphere size. Bolster et al. [20] employed similarity theory to find empirical relationships between the non-dimensional parameters that characterize a physical system. Qing et al. [21] conducted a two-dimensional numerical simulation of dynamics of free-surface Stokes flow dynamics.

This study built a two-dimensional simulation of the spreading dynamics of a single droplet 200 μm in diameter impacting a spherical surface 600 μm in diameter, and explored the effects of impact velocity, viscosity, surface tension, and surface size on the deposition of the droplet.

Experimental

The piezoelectric nozzle used in this experiment is about 10 μm in diameter and sprays droplets 20 μm in diameter (Fig. 1). However, according to the actual experimental conditions of the vast majority of research facilities, droplet modeling is limited to a droplet of 200 μm or larger in diameter impacting a spherical surface 600 μm in diameter. This study used similarity theory to establish empirical relationships between non-dimensional parameters, and conducted experiments under the same collision conditions (droplet 200 μm in diameter, spherical surface 600 μm in diameter).

Fig. 1

Microscopic image of inkjet nozzle of 3D printing bone scaffold, a single nozzle microscopic image under ×500 amplification; b microscopic image of multi nozzle array under ×100 amplification

Droplet parameters in piezo injection experiments

As shown in Fig. 2, the droplet spray system consists of a pulse function generator and homemade piezoelectric nozzle. The piezoelectric droplet injection process is very complex. Impact events were observed with a high-speed video camera (Photron Fastcam SA4) with a typical frame rate of 5400 frames per second (fps) and corresponding pixel resolution of 1024 × 1024 pixels. No significant information loss was observed between 1000 and 2000 fps for drop impact velocities up to 1.5 m s⁻¹. Data for lower values of impact velocity (1.0 m s⁻¹) were recorded at 1000, and 2000 fps recording speed was used for high values of impact velocity (1.5 m s⁻¹). The camera, which was mounted on a boom stand, captured 1000 fps in full frame (2764 × 2048) and 2000 fps with a reduced matrix resolution (1382 × 1024). All video images were saved directly to a computer for subsequent analysis. The microscope (Optem Zoom 65) used has a high-power (over 8 W) stroboscopic light source (Nova-Strobe DB Plus Digital), offering a maximum working distance of 110 mm and an extended zoom range for continuous coverage of 0.7–4.5 times. The microscope captured biobinder drop impact and the spreading process on the HA small ball surface.

Fig. 2

Bioadhesive microdroplet piezoelectric injection experiment schematic diagram. a Piezoelectric nozzle, b power and pulse function generator, c CCD high-speed camera, d light source, e spherical substrate, f biobinder droplet, g movable bracket, h computer

Bioadhesive was used as the medium in the homemade piezoelectric nozzle to create spray droplets. The droplet diameter was about 200 μm. A square wave signal was generated by a 20 Hz frequency pulse function. The piezoelectric nozzle was stimulated to generate a 200 μm bioadhesive drop which impacted the microspherial surface of the 600 μm HA microsphere. A sustained high-brightness light source which guaranteed the background was clearly visible during experimentation.

The velocity and diameter of the binder droplet were calculated according to the following method. As shown in Fig. 3, nozzle diameter l is a known value, and bioadhesive drop flying distance is represented by (s + d/2). d is the diameter of the biobinder droplet. Two variables were calculated by image processing software: the ratio of bioadhesive drop flying distance and nozzle diameter [a=(s + d/2)/l], and the ratio of nozzle diameter and droplet diameter (b = d/l). Thus, the velocity and diameter of the droplet can be calculated according to the following formulas:

Fig. 3

Bioadhesive droplet generated by piezo injection according to experimental schematic (Fig. 2) in an ordinary room

$$ \nu = (a_{2} l - a_{1} l)/(t_{2} - t_{1} ), $$

(1)

$$ d = b_{1} l\;{\text{or }}d = b_{2} l, $$

(2)

where ν is the droplet’s flying speed, d is droplet diameter, and t ₁ and t ₂ are different times. The ratio a ₁ and a ₂ corresponds to t ₁ and t ₂, the ratio b ₁ and b ₂ corresponds to t ₁ and t ₂, and l is the outer diameter of the nozzle.

In the experiments, a bioadhesive droplet (200 μm in diameter) collided with a spherical surface (diameter in 600 μm) at an initial velocity of 1.0 and 1.5 m s⁻¹. Both the contact angles were 90°. The diameter of the bioadhesive droplet was similarly defined as 200 μm as it impacted the microspherial surface of HA particles during the numerical simulation process. Particle size was also set to 600 μm to investigate droplet impact phenomena, (which is difficult to capture axisymmetrically in practice, especially at higher Weber numbers.) Impact processes were captured by the above system as shown in Fig. 3.

Governing equation of fluid flow numerical simulation

The isothermal impact of a liquid droplet on a spherical surface is a non-steady process that involves the liquid droplet and air. It is assumed that there is no mass or heat transfer between the two phases. 2D models in Cartesian coordinates comprised of continuity, momentum, and energy equations were solved as follows:

$$ \frac{\partial \rho }{\partial t} + \nabla \times (\rho \overrightarrow {v} ) = 0, $$

(3)

$$ \rho \frac{{{\text{d}}\overrightarrow {v} }}{{{\text{d}}t}} = - \nabla p + \mu \Delta \overrightarrow {v} + \frac{\mu }{3}\nabla \left( {\nabla \times \overrightarrow {v} } \right) + \overrightarrow {F}, $$

(4)

$$ \frac{{\partial \left( {\rho E} \right)}}{\partial t} + \nabla \times \left( {\overrightarrow {v} (\rho E + P)} \right) = \nabla \times (\kappa \nabla T) + S_{h}, $$

(5)

where $\rho$ is the density, $t$ is the time, $\overrightarrow {v}$ is the velocity vector, $p$ is pressure, $\mu$ is dynamic viscosity, T is temperature, $\overrightarrow {F}$ is the body force, $\kappa$ is thermal conductivity, and $S_{\text{h}}$ is the source of energy per unit volume per unit time.

In this study, the VOF method was used to obtain solutions. A single energy equation was solved for the mixture phase, the properties (ϕ) of which were proposed by Shen et al. [22]:

$$ \phi = \frac{{\sum {\alpha_{k} \rho_{k} \phi_{k} } }}{{\sum {\alpha_{k} \rho_{k} } }}, $$

(6)

where α _k is the volume fraction of the kth fluid droplet, and ρ _k is the density of the kth fluid droplet.

The interface between phases was tracked for each computational cell by computing the volume fraction of the fluid k:

$$ \frac{{\partial \alpha_{k} }}{\partial t} + V_{k} \times \nabla \alpha_{k} = 0. $$

(7)

Surface tension is a significant factor of fluids with non-zero contact angle on the surface. This effect can be incorporated into the momentum model by expressing it as an extra body force using the divergence theorem.

Initial/boundary conditions of numerical simulation

The effect of air is considered negligible due to its significantly smaller density and viscosity compared to the bioadhesive liquid, so the droplet surface is assumed to be a free surface. Fluent 6.3 was adopted to simulate biobinder droplet spraying behavior on the surface of the HA microsphere. As shown in Fig. 4, a droplet with an initial diameter D ₀ impacts a spherical surface with a diameter of D _S at a velocity of v ₀. The computational domain for 2D simulation is presented in Fig. 4. The domain dimensions of the mesh were determined based on the experimental observations of droplet maximum spread diameter. A domain size of 1.4 × 0.4 mm² was used for simulation, discretized by quadrilateral meshes of about 121,538 grids. Minimum grid spacing was 2 μm and maximum aspect ratio was 10 μm. Cells were dense near the spherical surface and grew gradually coarser toward the boundaries, effectively resolving the interface near the solid surface. The air–solid interface was considered to be a non-slip boundary. Contact angle varies in a narrow range during spreading and retraction according to Francois et al. [23], so here, it was set as a constant of α = 90°. A total of 4438 cells were patched for the biobinder droplet during simulation. A residual of 10⁻⁴ was set for convergence of continuity, and momentum and volume fraction equations were performed using a time step of 10⁻⁶ with 20–30 iterations per time step. A first-order implicit time stepping approach was adopted.

Fig. 4

Schematic of computational domain for microdroplet impacts on the hydroxyapatite particle

Results and discussion

In this study, spread diameter $d$ is defined as the diameter of liquid lamella on the spherical surface, and spread thickness $h$ as the thickness of liquid lamella at the central axis. Both parameters are critical in evaluating the droplet deposition on the hydroxyapatite spherical surface, and are non-dimensionalized with respect to the initial diameter D ₀. The resulting spread factor $\varepsilon$ and dimensionless spread thickness $h^{*}$ are as follows:

$$ \varepsilon \,= \,d/D_{ 0}, $$

(8)

$$ h^{*} { = }\;h/D_{0}. $$

(9)

Numerically simulated spreading process of biobinder droplet

Figure 5 shows the CFD simulation of droplet deformation with static contact angle as the wall boundary. The simulations fully captured the dynamics of the system using dynamic contact angle, interpreting the real impact of droplets on spherical surfaces. The system can be decomposed into four distinct stages: kinetic, spreading, relaxation, and equilibrium. Each stage is governed by the interaction of inertia, viscous force, and surface tension. The computer simulation process allowed prototyping fabrication parameters to be deduced rapidly.

Fig. 5

CFD simulation using dynamic contact angle at two different spraying speeds, a bioadhesive droplet spraying velocity of 1 m s⁻¹; b droplet spray velocity of 1.5 m s⁻¹

Kinetic phase

The droplet is assumed to be spherical in shape during the kinetic phase (first 10 μs in Fig. 5). A shock wave drives the oscillation that separates the compressed region at the lower part of the droplet from the non-compressed region at the upper part of the droplet. The upper part moves at a uniform velocity of v ₀, and is not affected by spherical surface properties, while the compressed liquid begins spreading out over the surface. The spreading is completely dictated by inertia during this process, and the effects of surface tension and viscous forces are negligible. The inviscid theory is adopted in this case.

Spreading phase

The spreading phase extends from the deformation of radially expanding lamella ejected from the base of the droplet to the maximum spread diameter at about 200 μs. Inertia is the driving force of spreading, whereas surface tension and viscous force provide resisting force according to Bakshi et al. [16]. The inertial effect dominates the early spreading of the droplet, and spread thickness changes in a linear fashion (Table 1); inertia is then partially or even fully dissipated, especially when the contact line velocity is close to 0, and surface tension and viscous forces determine the spreading of the droplet.

Table 1 Droplet spread thickness for different impact velocities during the first 50 μs

Relaxation phase

The droplet will retract once it reaches maximum diameter according to the stored elastic energy provided by surface tension. Viscous forces oppose elastic forces.

Equilibrium phase

The droplet is finally brought to an equilibrium state once oscillations cease. After 500 μs, the spread diameter of the lamella remains largely unchanged, but the surface oscillates slightly. The kinetic energy of the droplet is then dissipated by viscous forces or converted into surface energy.

Simulation was conducted under the same collision conditions described above. The diameter of the solid sphere was set to 600 μm for all numerical calculations. Droplet shapes which varied (see images in Fig. 5) at various times were reproduced accurately through simulation consistently throughout the entire collision process. Agreement between the simulated model and experimental model further verify the accuracy of the method proposed in this study—simulation results can, in fact, be used to characterize droplet impacts on spherical surfaces.

Effect of impact velocity on droplet deposition

The spreading of a droplet (D ₀ = 200 μm) impacting a spherical surface (D _s = 600 μm) was simulated at initial velocities of 0.5, 1.0, 1.25, and 1.5 m s⁻¹. In the liquid phase, density ρ _l was 990 kg m⁻³, kinematic viscosity coefficient μ _l was 0.0009 Pa s, and surface tension coefficient σ was 0.069 N m⁻¹. In the gas phase, density ρ _g was 1.215 kg m⁻³, and dynamic viscosity coefficient μ _g was 1.6105 × 10 Pa s. Weber numbers (W _e = ρ _l v ²₀ D ₀ /σ), from smallest to largest, were 0.68, 2.73, 4.27, and 6.15, and Reynolds numbers (R _e = ρ _l v ₀ D ₀ /μ _l) were 99.82, 199.64, 249.55, and 449.19, respectively. Weber number and Reynolds number indicate the impact energy of the droplet; both were observed to be small, indicating that the impact energy was low.

Figure 6 shows the temporal evolution of the droplet spread factor for different impact velocities. Maximum spread factor was observed at about 200 μs, at which point droplets reached maximum diameter of 1.43–2.20 times their initial diameters. Maximum spread diameter also increased as impact velocity increased, because the kinetic energy of a droplet is partly dissipated by viscous forces and partly converted into surface energy during the spreading phase; thus, the greater the impact velocity, the larger the converted surface energy and the larger the surface area and diameter of the resultant lamella. As impact velocity increased from 0.5 to 1.5 m s⁻¹, the time needed for the droplet to reach maximum spreading decreased slightly, from 230 to 210 μs. The curves flattened, with small fluctuations after 500 μs, and the final spread factor decreased from 1.43 to 1.23 as impact velocity increased from 0.5 to 1.25 m s⁻¹ before, surprisingly, increasing significantly (up to 2.02) with further increase in impact velocity (up to 1.5 m s⁻¹). This dramatic increase in spread factor can be attributed to the droplet breaking into smaller droplets at high impact velocity (1.5 m s⁻¹), as shown in Fig. 4a, b, which resulted in loss of surface tension energy. The lamella was thus unable to retract, creating large final spread diameter. (Spread diameter was defined as the external diameter of the lamella in this case.)

Fig. 6

Temporal evolution of spread factor for different impact velocities

Figure 7 shows the temporal evolution of the dimensionless spread thickness of the droplet for different impact velocities. Dimensionless spread thickness first decreases in an approximately linear manner, where a higher decrease rate is to be expected at higher impact velocity. Minimum thickness was 150–250 μs. The higher the impact velocity, the thinner the minimum spread. Spread thickness neared 0 at velocity of 1.5 m s⁻¹, indicating where the droplet broke. Spread thickness remained largely unchanged after 500 μs, however.

Fig. 7

Temporal evolution of dimensionless spread thickness for different impact velocities

Figure 8 shows spread thickness h in the first 50 μs plotted against time. Spread thickness decreased linearly, and the absolute value of the slope increased as impact velocity increased. The data in Table 1 were fitted with a linear function—let the slope of the linear function be $k_{{v_{0} }}$, where v ₀ is the velocity of the droplet, then

Fig. 8

Droplet spread thickness during the first 50 μs

$$ h = 200 - k_{{v_{0} }} t. $$

(10)

According to Table 1, it is assumed that $k_{{v_{0} }}$ equals v ₀, thus

$$ h = 200 - v_{0} t. $$

(11)

Spread thickness is completely determined by impact velocity during the initial collision, and the effect of the spherical surface is negligible. The upper part of the droplet falls as a rigid body, whereas the lower part is compressed and spreads outward. Table 1 shows that better agreement between calculated and simulated results was obtained at high impact velocity, indicating that spread thickness is less likely affected by other factors in this case. The difference between calculated and simulated results grows with time, because impact velocity decreases gradually with the dissipation of inertial energy and the role of other factors, such as surface tension and viscous forces, becomes more prominent according to Bakshi et al. [16].

Effect of curvature on droplet deposition

A two-dimensional simulation of the spreading of a droplet (D ₀ = 200 μm) impacting spherical surfaces with diameters of 400, 600, 800, 1000, and 10⁵ μm at an initial velocity of 1.0 m s⁻¹ was performed.

Figure 9 shows where during the initial collision no significant effect on spread factor the curvature had. Maximum spread factor at about 200 μs increased from 1.57 to 2.19 as the curvature increased from 400 to 10⁵ μm. Curvature likewise had no effect on the time needed for the droplet to reach maximum diameter. Once the droplet stopped spreading and reached equilibrium, spread factor increased significantly alongside increasing curvature.

Fig. 9

Spread factors of a droplet impacting spherical surfaces with different diameters

Figure 10 shows that during the initial collision, curvature had no significant effect on spread thickness. This further validates the conclusion drawn in the previous section that spread thickness decreases linearly with the v ₀ slope, and is less affected by the properties of the spherical surface. The smaller the curvature, though, the thinner the minimum spread thickness; thus, when the curvature is small enough, the droplet breaks at the center. During the equilibrium phase, the spread thickness decreased slightly with increasing curvature.

Fig. 10

Dimensionless spread thickness of droplet impacting spherical surfaces with different diameters

When D _s = 100 μm, the collision can be considered to occur between the droplet and a flat surface, due to the significant difference in size between them. Pasandideh et al. [24] provided a simple analytical model to predict maximum droplet size after impact, and predications were well in accordance with experimental measurements reported in previous research (at error within 15 %).

Maximum spread factor can be expressed as follows:

$$ \varepsilon_{\hbox{max} } = \sqrt {\frac{{W_{\text{e}} + 12}}{{3(1 - \cos \theta_{a} ) + 4(W_{\text{e}} /\sqrt {\text{R}_{\text{e}} } )}}}. $$

(12)

In this study, the theoretical value of maximum spread factor was 1.98, which had 9.59 % difference from the simulation value (2.19), confirming the reliability of the simulation results.

Effect of viscosity on droplet deposition

The spreading of a droplet (D ₀ = 200 μm) impacting a spherical surface (D _s = 600 μm) at an initial velocity of 1.0 m s⁻¹ under different viscosity coefficients (μ = 0.001, 0.002, 0.003, 0.005, and 0.02 Pa s) was simulated.

Figure 11 shows the temporal evolution of the droplet spread factor for different viscosity coefficients. Maximum spread factor, as well as the time needed for the droplet to reach its maximum diameter, decreased with increasing viscosity. 200 μs was required to reach maximum diameter when $\mu$ = 0.003 Pa s, 180 μs when $\mu$ = 0.005 Pa s, and 150 μs was needed when μ = 0.02 Pa s. This phenomenon can be attributed to the increase in viscous (resistance) force, while the droplet spread with increased viscosity coefficient.

Fig. 11

Droplet spread factor for different viscosities

Figure 11 also shows that when the viscosity coefficient was less than 0.003 Pa s, the spread factor increased alongside the increasing viscosity coefficient. This occurred because the increase in viscous force reduced the retraction of the lamella, thus resulting in larger spread diameter. When viscosity coefficient was larger than 0.003 Pa s, however, the final spread diameter decreased with increasing viscosity coefficient—the maximum spread diameter was small when μ = 0.02 Pa s, thus the final diameter was also small. For instance, when μ = 0.02 Pa s, the maximum spread factor was even smaller than the final spread diameter for other viscosity coefficients. The difference between maximum spread diameter and the spread diameter after the first retraction reflects the retraction ability of the droplet. The greater the viscosity coefficient, the greater the viscous force, and thus the smaller the difference and the poorer the retraction ability of the lamella.

Figure 12 shows that the curves of spread thickness against time for different viscosities coincide before 150 μs, indicating that viscosity had almost no effect on spread thickness during initial spreading. Minimum spread thickness increased as viscosity coefficient increased, however. At viscosity coefficients larger than 0.003 Pa s, the final spread thickness increased with viscosity.

Fig. 12

Dimensionless spread thickness of droplet for different viscosities

Effect of surface tension on droplet deposition

The spreading of a droplet (D ₀ = 200 μm) impacting a spherical surface (D _s = 600 μm) at an initial velocity of 1.0 m s⁻¹ at different surface tensions (σ = 0.03, 0.05, and 0.073 N m⁻¹) was simulated.

Figure 13 shows the temporal evolution of the droplet spread factor under different surface tensions. Surface tension had almost no effect on the spread factor at first, and thus is negligible compared to the inertia; however, it is important to note that surface tension had a significant effect on the time needed for the droplet to reach its maximum spread factor. For instance, the time needed for the droplet to reach maximum spreading decreased from 350 to about 250 μs, and the maximum spread factor decreased from 2.23 to 1.82 as surface tension increased from 0.03 to 0.073 N m⁻¹. As surface tension increased, the resistance force during droplet spreading increased, and kinetic energy was rapidly converted into surface energy. Conversely, during the equilibrium phase, spread factor decreased significantly as the surface tension increased from 0.03 to 0.05 N m⁻¹, which can be attributed to the droplet breaking and resulting in lost surface tension energy. As surface tension increased, maximum spread diameter decreased and the retraction ability of the lamella increased. Thus, the diameter of the lamella decreased after retraction.

Fig. 13

Droplet spread factor for different surface tensions

Figure 14 also shows that surface tension had almost no effect on dimensionless spread thickness at first. Minimum dimensionless spread thickness, as well as the dimensionless spread thickness during the equilibrium phase, increased with increasing surface tension, however. Surface tension thus shows a considerable effect on the spreading of a droplet impacting a spherical surface, especially as far as the time needed for the droplet to reach maximum spreading. As surface tension increases, maximum spread factor (as well as the time needed for the droplet to reach maximum diameter) decreases. Surface tension occurs in the direction opposite inertia, and thus inhibits the spreading of the droplet; however, it allows the retraction of the droplet once it reaches maximum diameter.

Fig. 14

Dimensionless spread thickness of droplet for different surface tensions

Conclusion

The most notable conclusions of this study can be summarized as follows:

• (1) As impact velocity increases, the maximum spread diameter of the droplet increases, minimum spread thickness decreases, final spread diameter decreases, and final spread thickness increases. As curvature increases, maximum spread diameter increases, minimum spread thickness increases, final spread diameter increases, and final spread thickness decreases. As viscosity coefficient increases, maximum spread diameter decreases, minimum spread thickness increases, final spread diameter decreases, and final spread thickness increases. As surface tension coefficient increases, maximum spread diameter decreases, minimum spread thickness increases, final spread diameter decreases, and final spread thickness increases.

• (2) Impact velocity, curvature, and viscosity coefficient have no significant effect on the time needed for the droplet to reach maximum spreading, but surface tension has a significant effect. Time needed decreases from 350 to about 250 μs as surface tension increases from 0.03 to 0.073 N m⁻¹.

• (3) Dimensionless spread thickness first decreases in an approximately linear manner, and the rate of decrease is completely dictated by the impact velocity, whereas the effects of curvature, viscosity coefficient, and surface tension are negligible.

• (4) When impact velocity is too high or the curvature of microspheres or biobinder surface tension is too small, the bioadhesive droplet breaks up into smaller droplets. Air at the liquid–solid interface is trapped at the moment of impact if the viscosity coefficient is too large.