Nanotechnology has been presenting successful applications in several fields, such as electronics, medicine, energy, and new materials. However, the high cost of investment in facilities, equipment, and materials as well as the lack of some experimental analysis at the nanoscale can limit research in nanotechnology. The implementation of accurate computer models can alleviate this problem. This research investigates the Leidenfrost effect at the nanoscale using molecular dynamics (MDs) simulation. Models of water droplets with diameters of 4 nm and 10 nm were simulated over gold and silicon substrates. To induce the Leidenfrost effect, droplets at 293 K were deposited on heated substrates at 373 K. As a baseline, simulations were run with substrates at room temperature (293 K). Results show that for substrates at 293 K, the 4 nm droplet has higher position variability than the 10 nm droplets. In addition, for substrates at 373 K, the 4 nm droplets have higher velocities than the 10 nm droplets. The wettability of the substrate also influences the Leidenfrost effect. Droplets over the gold substrate, which has hydrophobic characteristics, have higher velocities as compared to droplets over silicon that has a hydrophilic behavior. Moreover, the Leidenfrost effect was observed at the boiling temperature of water (373 K) which is a significantly lower temperature than reported in previous experiments at the microscale. This research lays the foundation for investigating the fluid–structure interaction within several droplet based micro- and nano-manufacturing processes.
Nanotechnology has been presenting several applications despite being a recent field. Applications include compact transistors that make faster and energy efficient processors and memory chips , long-life battery cells , efficient drug-delivery systems , DNA sequencing chips , stronger structural materials , superconducting materials , and others. These successful applications of components measuring less than 100 nm have demonstrated the potential of nanotechnology and the need for a better understanding of material properties at the nanoscale. Working with nanotechnology often requires expensive materials, specialized equipment, and state-of-the-art facilities. Besides the high cost of these resources, they also require long setup times, trained personnel, and complex procedures. The high cost of procurement and operation of nanotechnology resources can restrict the development in this field. To the best of our knowledge, there is no microscopy method that can observe droplet at the nanoscale, without influencing the measurements of the experiments. One solution to help optimize the R&D in nanotechnology is to use computer models to help optimize design and process parameters of the experiments and products, hence reducing costs and increasing design freedom.
Traditional numerical simulation methods, such as finite elements, predict material properties up to the submicron scale [7,8]. However, they fail to make correct predictions for components smaller than several hundred nanometers . A more accurate approach for computer modeling for the nanoscale is to use molecular dynamics models. In MDs models, each atom in the system and their interaction with other atoms are represented. The representation accuracy, freedom of design, and insights that can be gained by using MD modeling have attracted several research groups to investigate nanotechnology applications. Tada et al.  used MD simulation to investigate the failure mechanism of Si molds and reduce defective outputs of nano-imprint lithography (NIL) process. Chen et al.  used a hybrid approach of MD simulation and experiments and proposed a highly sensitive sensor to detect molecular conformation. Desai et al.  used MD simulations to investigate the wettability of and as a function of temperature. Borodin et al.  used MD simulations to investigate the properties of lithium batteries and minimize interfacial resistance, improving battery safety and battery life.
This research demonstrates the applicability of molecular dynamics to investigate the Leidenfrost effect at the nanoscale which is an underexplored phenomenon. Several direct-write droplet based manufacturing processes involve the deposition of micro- and nano-scale droplets on heated substrates. These include scalable inkjet [14,15], aerosol jet , electrohydrodynamic jet , and other atomized droplet processes where the droplet deposition dynamics determines the morphology of the printed feature. Thus, it is critical that the transport properties of droplet be investigated in order to determine the fluid–structure interaction between different substrate materials at elevated temperatures. In addition, this research investigates the droplet movement on heated substrates, which has an impact on the final placement and stabilization shape of the droplet. The phenomena of water interfacing with a hot surface were first investigated by Leidenfrost in 1756 . If a droplet of liquid is deposited over a surface around the boiling temperature of the liquid, the liquid boils and evaporates rapidly. However, when the temperature of the surface is significantly higher than the boiling temperature of the liquid, the droplet levitates over its own vapor, greatly increasing the evaporation time of the droplet. Also, the thin layer of vapor avoids the nucleation of bubbles, preventing the droplet from boiling and making it evaporate slowly . This research investigates the Leidenfrost effect at the nanoscale based on variations in droplet size, substrate temperature, and substrate material. We track the droplet trajectory path via its centroid and its velocity overtime. The tools, parameters, and models used in this work can be extended to other materials and substrate geometries to enable other scenarios of the Leidenfrost effect to be studied. The results of this research provide quantitative analysis on the role of different process parameters that impact the Leidenfrost effect at the nanoscale regime. Further, they provide a basis to investigate several droplet based nano- and micro-manufacturing processes.
In this research, the nanoscale molecular dynamics (NAMDs)  source code was used in conjunction with virtual molecular dynamics (VMDs) software to model the materials in the system and visualize the simulation output. NAMD is robust for parallel computing [20,21] and open source and compatible with most force fields for commonly available CHARMM  and AMBER  packages. The MD simulations were executed on a graphical processing unit (GPU) processor (NVIDIA Tesla K40) with 2880 CUDA cores on a 64-bit Linux based system to enhance the performance of the computations . The NAMD source code uses the GPU for nonbonded force evaluation while the energy evaluation is done on the central processing unit .
In this work, MD models of 4 nm and 10 nm water droplets were used, containing 1108 and 17,267 molecules, respectively. Substrates of gold and silicon measuring 240 Å × 240 Å × 40 Å in dimensions were used. The water molecules had a TIP3P structure and were modeled using VMD. All the force fields used in the simulations are compatible with the CHARMM standard, and the format of potential energy function shown in Eq. (1) was used to represent the atomic interactions. The simulations were performed with a 2 fs integration time step. Van der Waals interactions were computed with a cutoff of 12 Å and switching function starting at 10 Å. The long-range electrostatic forces were computed using a particle mesh Ewald (PME) summation method. A canonical ensemble of conserved number of atoms, volume and temperature was used, and a Langevin thermostat was employed to control the temperature. The water droplets were modeled as spheres and placed on top of the substrates, as illustrated in Fig. 1
The parameters for Eqs. (2)–(5) for water molecules (oxygen and hydrogen) were taken from the CHARMM force fields . The parameters for gold and Si substrates were adapted from Braun et al.  and Mayo et al. , respectively, as shown in Table 1.
The objective of the simulations was to analyze the Leidenfrost effect which describes the fluid–substrate interactions at the nanoscale. An important aspect of this research was to investigate the variations in the droplet velocity over the substrate as a function of the substrate material and temperature. The trajectory path of the droplet was used to analyze the relationship between the droplet movement and the physics of the Leidenfrost phenomena. The centroid of the droplet was recorded every 2 ps to construct a trajectory path. The velocity of the droplet was computed by differentiating the droplet positions over time. Scripts in VMD were used to compute the droplet position, and matlab was used to calculate the velocities. Two temperature levels were used in the simulation: 293 K and 373 K. Table 2 shows the design of experiments based on the combination of droplet size, substrate material, and temperature. Simulation runs were conducted to execute the three factors (k) at two levels each (2 k = 23 = 8 runs).
The behavior of nanoscale water droplets coming in contact with a heated substrate was simulated. The simulations run at 293 K served as a base line to observe droplet behavior at room temperature. In order to simulate a heated substrate, the entire simulation volume was initially thermalized for 5 ps, wherein the system reaches a temperature of 293 K. After this initial phase, the substrate was heated and maintained at 373 K using a Langevin temperature control thermostat for the entire simulation period.
Results and Discussion
Nanodroplets at 4 nm and 10 nm were tracked using their trajectory profiles. The effect of droplet size, substrate material, and temperature on the nanodroplet spreading dynamics was observed over time. Further, the presence of the Leidenfrost effect was verified by tracking the kinetic energy and temperature of different layers of the nanodroplet from the substrate surface.
One of the first evidence of the Leidenfrost effect on a droplet over a flat surface is that the droplet tends to move over the substrate. Figure 2 shows the top view of 4 nm and 10 nm droplets deposited over gold and silicon at 293 K after 2 ns of simulation.
It can be seen that the droplets deposited over a substrate at room temperature (293 K) have minimal displacement within the 2 ns time frame. The droplets have a z-directional displacement when they spread over the substrates (see Fig. 10), but remain stationary in x–y directions. The water nanodroplets display both hydrophilic and hydrophobic interactions with respect to silicon and gold substrates (Fig. 10). The trajectory path of the droplets (bold line) remains at the center of the simulation volume, indicating that the droplets maintain a steady position. However, in contrast, a droplet deposited over a heated substrate (373 K) starts to move in random directions. Figure 3 shows the top view of 4 nm droplets moving over gold (a) and silicon (b) substrates over time at 373 K.
From Fig. 3, it is evident that after 1 ns the droplets moved to a different region on the substrate. Droplets deposited over gold have a higher displacement than droplets deposited over silicon. After 2 ns, the 4 nm droplet presents a significant displacement and is positioned at the edge of the substrate. However, the 4 nm droplet over silicon moved significantly less than the droplet over gold. To better illustrate the movement of the droplets, the trajectory path of 4 nm and 10 nm droplets deposited over gold and silicon substrates at 373 K is shown in Fig. 4.
The 4 nm droplets over gold at 373 K present the highest displacement and its trajectory path is the longest. Ten nanometer droplets over gold present a significant displacement, but smaller than the 4 nm droplet. Nanodroplets over silicon present a lower displacement as compared to droplets over gold. Similar to the gold substrate, the 4 nm droplets show higher displacement as compared to the 10 nm droplet over the silicon substrate. This is due to the fact that larger nanodroplets have higher aggregate mass which limits its mobility based on the vapor layer formation in between the droplet and the substrate.
According to Leidenfrost , droplets move on a heated substrate due to a thin layer of vapor that forms between the droplet and the heated surface. Typically, the vapor layer separates the droplet from the surface and propels it in random directions, depending on the flow direction of the vapor. Experiments at the microscale have demonstrated that the Leidenfrost effect occurs at temperatures above 473 K . One important finding from our simulations with nanoscale droplet reveals that this effect was observed at 373 K. This temperature to observe the Leidenfrost effects is significantly lower than temperatures at the microscale. We further explore this effect by dividing the droplets in layers of 0.5 Å in height and measuring the kinetic energy of molecules in each layer away from the substrate surface. Figures 5(a) and 6(a) show the average kinetic energy of the atoms as a function of the distance away from the gold substrate surface for 4 nm and 10 nm droplets, respectively. It can be observed that at 373 K (solid line legend), nanodroplets display higher molecular kinetic energy for layers closer to the substrate. However, there is a steep decrease in the kinetic energy of molecules for distance over ∼4 Å from the substrate. The graphs at 373 K for both 4 nm and 10 nm droplets depict the presence of two distinct water phases in the simulation. This includes a thin layer of vapor next to the substrate with a higher kinetic energy and a liquid phase in the layers further away from the substrate with lower kinetic energy. In addition, the 4 nm droplet has higher kinetic energy (9750 kcal/mol) as compared to the 10 nm droplet (8300 kcal/mol). This can be attributed to the lower escape velocities of the vapor molecules from under the nanodroplet for larger droplet sizes due to higher aggregation of molecules for the 10 nm droplet. In contrast, when nanodroplets are placed over a substrate at 293 K (dashed line), the molecular kinetic energy is similar for all the layer of molecules. Thus, this constant kinetic energy (2000 kcal/mol) illustrates that the droplet has only one water state, which is liquid. The main characteristic of the Leidenfrost effect is the fact that droplets undergoing this phenomenon require a longer time period for evaporation when placed on a heated substrate. This is because a vapor boundary layer is formed between the substrate and the lower layers of the droplet interfacing with the substrate. The presence of the vapor layer reduces the heat exchange between the substrate and the nanodroplet, resulting in longer evaporation times for the droplet. Figures 5(b) and 6(b) show the temperature profiles of the atoms as a function of the distance away from the gold substrate surface for 4 nm and 10 nm droplets, respectively. The temperature profiles are analogous to the kinetic energy profiles and display a steep rise in the temperature of the atoms in the vicinity of the substrate. This steep increase in temperature can be attributed to the formation of superheated steam at the droplet and substrate interface. The rapid escape of these high energy vapor molecules from the bottom of the droplet causes it to propel the droplet in the lateral (x–y plane) direction. It is important to note that the temperature and kinetic energies of atoms above the vapor boundary layer are much lower as compared to the peak values. If the peak values in kinetic energy and temperature were primarily convection based, then the average kinetic energy in the subsequent layers of the droplet would also increase over time. This was not observed in our simulation results suggesting that phenomena being observed were in fact the Leidenfrost effect.
When the Leidenfrost effect occurs, the heat transfer between substrate and water droplet is significantly reduced. The Leidenfrost effect negatively impacts heat-transfer rates for applications which rely on convection-based heat transfer between liquid media and solid surface. Thus, in practical applications such as industrial coolers, chillers, computer chips and other devices, there is a steep drop in thermal heat transfer efficiency due to vapor formation based on the Leidenfrost effect. Our research plays an important role to investigate and explain the Leidenfrost effect for nanoscale liquid–solid interfaces to resolve heat-transfer issue in heat exchangers. In addition, this research investigates the fluid–structure interaction between nanodroplets and different substrate materials that exist in several droplet based nano- and micro-manufacturing processes.
The velocity of the droplet calculated based on its centroid (as represented in Eq. (6)) was used to infer about the vapor layer formation. The position and velocity measurements for the nanodroplets were conducted in a three-dimensional coordinate system. Figures 7–9 show the instantaneous velocities for nanodroplets at 293 K and 373 K on gold and silicon substrates, respectively. As can be seen in the graphs, there is an initial increase in the velocity which can be attributed to the z-directional component of the velocity. This occurs as the droplet undergoes a vertical displacement from its initial spherical shape to its stabilization shape on the substrate. This phenomenon is captured within 100–200 ns since the beginning of the simulation depending on the size of the droplet and type of substrate. After attaining equilibrium contact angle (Fig. 10), the droplet is propelled on the substrate due to the Leidenfrost effect in the x–y plane. Thus, the motion of the droplet in the lateral plane (x–y plane) is the primary component to the velocity as shown in Figs. 8 and 9. The higher the droplet velocity, the more intensified is the vapor formation process. The velocity of the droplets (4 nm and 10 nm) at room temperature was computed to serve as a base case to study the Leidenfrost effect. Figure 7 shows the velocities of 4 nm and 10 nm droplets over gold and silicon at 293 K. The velocities in Fig. 7 represent the random vibration of the droplet around its stationary mean position. These droplet velocities occur when the droplet stays in its original position and is reflected as “noise” in the system. The droplet velocity calculation based on its centroid position is calculated as shown in the following equation:
where is the three-dimensional droplet centroid position at time t, and is the step size of the simulation (2 fs).
Figure 7 shows that the stationary vibration velocities of the 4 nm droplets are higher as compared to 10 nm droplets. This can be explained by the fact that smaller droplets have lower number of molecules (1108). The velocity is based on the position of the droplet center of mass, which in turn is a weighted average of all the molecules positions in the droplet. The bigger the pool of molecular positions used to calculate the center of mass, the lower its position variability. Because the velocity of the droplets is calculated as the derivative of position, the droplet position variability can in turn be derived into velocity. In other words, the random variability noise of the molecular centroid could cause the derivative to be greater than zero, even with the droplet being stationary. In this way, the higher the pool of molecules, the lower the position variability of the droplet and therefore the lower its velocity. The 10 nm droplets have more molecules (17,267), and the individual droplet displacement is an average of the pool of its molecules. One can expect that as the number of molecules grows, the droplet velocity would reduce further, and eventually be close to zero for droplets at the mesoscale. The random vibration of stationary droplets around its mean position can be used as a baseline velocity to benchmark moving nanodroplets. The velocity of the nanodroplets subjected to the Leidenfrost effect has much higher velocities as compared to baseline velocities of stationary droplets. This is evident for the nanodroplet motion on both the substrates. The average baseline velocity of 10 nm droplet on silicon substrate at 293 K was around 1 m/s as compared to 4 m/s at 373 K due to the Leidenfrost effect. Similarly, the average baseline velocity of 4 nm droplet on silicon substrate at 293 K was around 3 m/s as compared to 6 m/s at 373 K due to the Leidenfrost effect. In case of the gold substrate, the average baseline velocity of the 10 nm droplet at 293 K was around 2 m/s as compared to 5 m/s at 373 K due to the Leidenfrost effect. The same fact applies to the 4 nm droplet which had an average baseline velocity of 4.5 m/s at 293 K as compared to 12.5 m/s at 373 K. Thus, the moving droplet velocities are much higher as compared to random vibration of stationary droplets which indicates the presence of the Leidenfrost effect at the nanoscale.
When the temperature of the substrate is increased to 373 K, the droplets undergo random motion along the substrate, representing the Leidenfrost effect. Figure 8 shows the velocity of 4 nm and 10 nm droplets over a gold substrate at 373 K. The simulation of 4 nm and 10 nm droplets was terminated at 2 ns and 5 ns, respectively. Smaller droplets have a faster stabilization time at room temperature (293 K). In addition, droplets deposited over a heated plate (373 K) display movement and velocity peaks in a shorter time as compared to 10 nm droplets.
As can be seen from Fig. 8, the 4 nm droplet has significantly higher velocity as compared to the 10 nm droplet. At 373 K, the average velocity of the 4 nm droplet is around 12.5 m/s as compared to 5 m/s for the 10 nm droplet on the gold substrate. Besides having fewer molecules that average for the center of mass, the smaller droplets also have a higher surface to volume ratio. Thus, a smaller droplet has more atoms on its surface, where they are subjected to fewer hydrogen bonds. This facilitates the breakage of hydrogen bonds and formation of vapor. The accelerated vapor formation increases the droplet velocity. Moreover, smaller droplets have lower inertia and a lower resistance to displacement. Similarly, higher migration velocities are observed for the 4 nm droplets when deposited over a silicon substrate at 373 K, comparing the 10 nm droplets deposited over the same substrates (shown in Fig. 9).
As can be seen from Fig. 9, the 4 nm droplet has higher the velocity (6 m/s) as compared to the 10 nm droplet (4 m/s) on the silicon substrate. However, both the 4 nm and 10 nm droplets maintain a higher velocity over gold as compared to silicon substrate. The reduced velocity of droplets over the silicon substrate is due to the fact that the interaction of water nanodroplets with silicon is more hydrophilic than gold. The higher affinity between water molecules and the silicon substrate implies that more energy is necessary to separate the water droplet from the substrate by a layer of vapor, a condition necessary for the Leidenfrost effect to occur. Figure 10 shows the side view and the contact angle for 10 nm droplets over gold and silicon.
The contact angle of the 10 nm droplet over gold is 125 deg clearly indicating the hydrophobic interaction of water and gold substrate. The contact angle of the 10 nm droplet over silicon is 70 deg, thus making a hydrophilic substrate. The silicon substrate slows down the droplets movement because of the higher adhesion between them. This phenomenon restricts the formation of vapor layer and limits its lateral motion on the substrate.
This research explores the Leidenfrost effect at nanoscale dimensions using molecular dynamics simulations. Water droplets at 4 nm and 10 nm were simulated over gold and silicon substrates at 293 K and 373 K, respectively. At 293 K, both the droplets remained stationary on the substrate limiting their displacement from their original position. However, smaller droplets (4 nm) displayed higher random velocity about their mean position as compared to the 10 nm droplets. This can be attributed to the fact that smaller droplets have fewer atoms enabling higher variability for its centroid due to the smaller aggregation pool. The Leidenfrost effect was observed when droplets were deposited on the heated substrate at 373 K. This result is in contrast with the Leidenfrost effect at the macro- and micro-scale which is typically observed for temperatures over 473 K. At 373 K, the 4 nm droplets presented higher propagation velocities than the 10 nm droplets. This is due to the fact that smaller droplets have a higher surface to volume ratio which makes the smaller droplets absorb higher energy per unit volume. Also, molecules at the surface of the droplet have fewer hydrogen bonds and require less energy to separate from other molecules. Thus, when exposed to a heated substrate, the breakage of hydrogen bonds is accelerated on smaller droplets as they possess proportionally fewer hydrogen bonds. In addition, the smaller droplets have a lower inertia and thereby are prominently influenced by the propelling forces of the vapor layer. Droplets deposited over gold substrate had higher velocities than droplets deposited over silicon. Silicon substrates are more hydrophilic than gold substrates, and the affinity between liquid and substrate acts as a restrain to the droplet movement. These results reveal the interplay of different process parameters which impact the Leidenfrost effect, an unexplored phenomenon at the nanoscale. This research forms a foundation to understand nanoscale droplet propagation and heat transfer within several droplet based nano- and micro-manufacturing processes.
The authors extend their gratitude to the U.S. National Science Foundation (NSF CMMI: Award No. 1435649) and the TMCF-Army Research Laboratory Faculty Research Fellowship for support toward this research.
bending constant of the bond
stretching constant of the bond
droplet centroid at time
charge of atom
distance between atom and
reference bond length (length with minimal stretching potential energy)
- t =
potential energy from bending interactions
potential energy from stretching interactions
potential energy from nonbonded interactions
total potential function
droplet velocity at time
time step between position measurements
depth of potential energy well
electromagnetic permittivity of free space
angular displacement of the bond
reference bond angular displacement
distance between atom and at which the interparticle potential is zero