Molecular Dynamics Simulation of Behaviours of Non-Polar Droplets Merging and Interactions with Hydrophobic Surfaces

Journal of Bionic Engineering 5 (2008) 271–281

Molecular Dynamics Simulation of Behaviours of Non-Polar Droplets Merging and Interactions with Hydrophobic Surfaces Y.Y. Yan, C.Y. Ji School of the Built Environment, University of Nottingham, Nottingham NG7 2RD, UK

Abstract This paper presents a molecular dynamics simulation of the behaviours of non-polar droplets merging and also the fluid molecules interacting with a hydrophobic surface. Such behaviours and transport phenomena are popular in general microchannel flow boiling and two-phase flow. The droplets are assumed to be composed of Lennards-Jones type molecules. Periodic boundary conditions are applied in three coordinate directions of a 3-D system, where there exist two liquid droplets and their vapour. The two droplets merge when they come within the prescribed small distance. The merging of two droplets apart from each other at different initial distances is tested and the possible larger (or critical) non-dimensional distance, in which droplets merging can occur, is discussed. The evolution of the merging process is simulated numerically by employing the Molecular Dynamics (MD) method. For interactions with hydrophobic solid wall, a system with fluid confined between two walls is used to study the wetting phenomena of fluid and solid wall. The results are compared with those of hydrophilic wall to show the unique characteristics of hydrophobic interactions by microscopic methods. Keywords: molecular dynamics simulation, mist flow, droplets merging, hydrophobic, wetting Copyright © 2008, Jilin University. Published by Elsevier Limited and Science Press. All rights reserved.

Nomenclature F: intermolecular force m: mass N: molecular number in the system r: intermolecular distance T: temperature V: system volume İ: energy parameter of L-J potential

1 Introduction These are common phenomena in many natural and practical processes that droplets or mists merge during the two-phase flow, and the fluid molecules also interact with surfaces of hydrophobic/hydrophilic properties. Such transport phenomena, which are very popular in general microchannel flow boiling and two-phase flow, and also often seen in process industry and other biological flow systems, have been studied by many scientists and engineers during the last few decades. The studies of such natural phenomena are often carried out Corresponding author: C.Y. Ji E-mail: [email protected]

ı: length parameter of L-J potential or surface tension I : potential energy

Subscripts L: liquid S: solid by using different methods, namely, experimental, theoretical analysis or numerical simulations. Nevertheless, for a specific physical problem, its intrinsic controlling mechanism is sole, no matter what type of method is applied, what kind of conceptual framework is employed, or what kind of superficial conclusions can be drawn. Previous numerical study and modelling of microchannel flow using conventional CFD method and Lattice Boltzmann Method (LBM) have been studied by many researchers, and this has been reviewed and reported by the authors recently[1,2]. On micro/mesos

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scale simulations, a LBM approach of the biomimetics of natural hydrophobic surfaces was reported[3,4]. Compared with the above, the Molecular Dynamics (MD) simulation method is a more effective tool for microscopic insight of complex physical phenomena. Also in comparison with experimental method the MD method bears the common advantages of numerical methods[5] such as a) Economy (low cost); b) Efficient (short period of investigation); c) Complete information (few inaccessible location and no disturbance caused by probes; hidden detail behind bulk measurements can be revealed[6]); d) Being able to simulate realistic conditions (little difficulty in having very small dimensions, treating very low or high temperatures, handling toxic or flammable substances, following very fast processes); e) Suitable for simulating ideal conditions (many idealizations such as two-dimensionality, constant density, an adiabatic surface, or infinite reaction rate can be easily and exactly set up). In particular, the MD method is suitable for simulating very small volumes of liquid flow, with linear dimensions of the order of 100 nm or less and for time intervals of several tens of nanoseconds. Therefore, it can deal effectively with nano-domains and is perhaps the only accurate approach in simulating flows involving very high shear stresses where the continuum or the Newtonian hypothesis may not be valid. For dimensions less than approximately ten molecules, the continuum hypothesis breaks down even for liquids, and MD should be employed to simulate the atomistic behaviour of such a system[7] to provide new insights as well as hard-toobtain quantitative information to complement other techniques. Therefore, there is a great potential for the MD method being applied to simulate such behaviours and transport phenomena in micro fluidics, microchannel flow boiling/two-phase flow, and biological mass flow such as blood flow in blood vessels. The MD simulation starts from a set of molecules occupying a region of space, with each assigned a random velocity corresponding to Boltzmann distribution at the temperature of interest. The interaction of the molecules is prescribed, commonly in

the form of a two-body potential energy, and the time evolution of the molecular positions is obtained by integrating Newton’s equations of motion. Based on the integration over time, the behaviours of the molecules such as the averages of density, velocity, stress, temperature, fields etc. can be calculated. The latter quantities may then be compared with continuum fluid information[6]. In order to avoid the calculation time scaling to O(N2) for N molecules, which is due to interactions between all pairs of molecules, the weak tail of the potential is commonly cut off at a distance of rc, and shifted by a term linear in r so that the force goes smoothly to zero at the cut-off. The MD simulation has long been used in statistical mechanics and chemistry. It is also used to study microscopic heat transfer phenomena and has started to be employed more widely in chemical process and pharmaceutical industries, typically in the analysis of the multiphase flow and micro-fluidic process. It is now widely used as a microscope in length scale and high-speed video recorder in time scale. The MD is appropriate because by using this technique the solid walls can be modelled explicitly. The decreasing tendency in length scale of microchannels of the cooling devices makes the MD simulation method to be a more and more important approach for predicting the characteristics of fluid flow and heat transfer. The main topics that the MD simulation may cover include thin film thermal conductivity, interfacial phenomena, phase change, transport coefficient, etc. The well-known model fluids used in MD include the “hard spheres fluid”, the “square well fluid” and the “Lennard-Jones” (L-J) fluid. The hard spheres fluid does not exhibit the transition of vapour-liquid phases because only repulsive interactions are present. The square well fluid, whose potential incorporates both repulsive and attractive forces between molecules but in a very simple way, is normally applied in simple atomic systems because of its simplicity and analytic tractability. On the other hand, the L-J fluid shows the transitions between vapour-liquid, solid-liquid and solid-vapour phases, and the critical and triple points. Therefore, the L-J fluid is well regarded as a suitable reference fluid for modelling properties of real fluids[8]. Other

Yan and Ji: Molecular Dynamics Simulation of Behaviours of Non-Polar Droplets Merging and Interactions with Hydrophobic Surfaces

intermolecular potentials of fluids, such as WeeksChandler-Andersen (WCA) potential, Buckingham potential, and Coulomb potential, have various restrictions for applications in comparison with the L-J potential[7]. The WCA potential is a modification of the L-J potential but can be used only in those cases when the atoms purely repel each other; the Buckingham potential is the one to be evaluated much more expensively; the Coulomb potential accounts for the electrostatic interactions between particles when charges take place, e.g., for ions or polyatomic molecules with partial charges. The L-J potential is written with the generalized “12-6” form of

Iij (rij ) 4H ij [cij (V ij / rij )12  dij (V ij / rij )6 ] .

(1)

The first term of the equation is an arbitrary but convenient short-range repulsion which prevents overlap of the atoms in space; while the second term represents the attractive polarised interaction of neutral spherical atoms. rij is the intermolecular separation distance between particles i and j. İij and ıij are the minimum energy and the zero energy separation distance relative to the pair, respectively. cij and dij are adjustable parameters which can be chosen to control the molecular interactions, for example, the wetting characteristic of fluid-solid interactions. For the distances smaller than ıij, the resulting force is repulsive; whereas it is attractive for larger distances. The L-J potential in Eq. (1) adequately describes the interactions between spherical non-polar particles, which include many mono-atomic fluids (noble gases such as Ar, Xe, Ne, Kr, etc.) and some small diatomic and polyatomic compounds (N2, O2, CO2, CH4, etc.). For polar fluids, the Coulomb part should be considered in their potential equation, which is not covered in the present study. Apart from pure non-polar fluids, L-J potential can also be used to simulate mixtures by applying the combining rules. In this case, due to its simple shape, the advantage of L-J potential is obvious because a two-combining-rules model is simpler to deal with than a three-or-more-combining-rules model[9]. The L-J potential is known to give a reasonably

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quantitative description of liquid argon, with parameters İ = 1.67×10í21 J, ı = 0.3405 nm, and c = d = 1. Long et al.[10] employed a MD method to simulate the complete evaporation of an argon droplet under subcritical conditions using 2048 particles. Consolini et al.[11] chose to examine the evaporation characteristics of Xe droplet in a N2 ambient because the Xe/N2 system allows droplet behaviour to be studied over a wide range of conditions, including subcritical, trans-critical, and supercritical pressures. In this study we investigate the interactions of two tiny droplets in an attempt to provide some information for the mist flow from a microscopic viewpoint. Effects of surface wettability on the behaviour of liquid atoms near a solid boundary were also studied by using MD simulation method[12,13]. The effect of an increased density near the wall was referred to as wetting of the surface by Markvoort et al.[14]; they used equal mass and size for both gas and solid particles, assuming the system realistic by corresponding to argon gas and calcium crystal. Thomas and McGaughey investigated the effect of surface wettability on liquid structure, mobility, and diffusion near a solid surface by applying MD simulation[13]. The density and structure of a liquid near a solid is dependant on the surface wettability. Near the more-wetting surface, both the liquid structure factor and density are well above bulk liquid values. Near the less-wetting surface, the liquid assumes enhanced structure but the density is low. The structure factor profiles suggest a natural division between a liquid region that is moderated by solid-liquid interactions and a liquid region that is moderated by liquid-liquid interactions. In the present study we carry out simulations for hydrophobic fluid-solid interactions with a comparison of hydrophilic interactions. Flow patterns in mini/micro- channels were observed typically as: dispersed bubble, bubbly flow, slug flow, churn flow, annular flow, and mist flow. Many interesting phenomena have been observed in experiments. In dispersed bubble pattern, numerous small bubbles float in a continuous liquid phase. In bubbly flow, bubble size is comparable to but not as large as the channel diameter. In slug flow, bubbles

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develop into bullet shape due to the channel wall restriction, sometimes the bullet bubbles are followed by a stream of small bubbles creating a trail. In churn flow, bullet bubbles start to distort and small bubbles in liquid slug coalescence into gas clump with the increase in gas velocity, which is a highly oscillatory flow with chaotic interface. In annular flow, gas phase becomes a continuous flow in the core of the channel. In mist flow, liquid film is blown away from channel wall and numerous liquid droplets float in high-speed vapour flow. The rupture and coalescence of droplets occur in mist stage of two-phase flow and boiling in microchannel. On the whole, the merging of two liquid droplets is a problem of continuum fluid mechanics, but the breaking of the interface is at molecular length and time scales. Therefore, MD simulations are applicable to reveal the detail of the microscopic dynamics of the merging process. The MD study is in particular important for the mist flow because it lacks direct observations for the characteristics of the notable amount of entrained liquid droplets. In the present study, two popular transport phenomena in general microchannel flow boiling and two-phase flow are simulated. One is the micro droplets or mists merging, the other is the fluid molecules interacting with the wall. Two identical droplets which are Lennard-Jones type fluid molecule clusters are assigned in one simulation system. The two droplets merge when they come within a prescribed small distance from each other. The merging process is simulated and presented in details at different time steps. For interactions with hydrophobic solid wall, a system with fluid confined between two walls is used to study the wetting phenomena of fluid and solid wall. The results are compared with those of hydrophilic wall to show the unique characteristics of hydrophobic interactions by microscopic methods.

2 Methodology Two different simulation systems are employed to investigate the droplet merging and interactions with hydrophobic walls, respectively. Common MD simulation techniques are used with special treatment for their own cases of the two systems.

2.1 Droplet merging In the present study, MD simulations based on the Nose method[15] of the canonical ensemble NVT (Number of molecules in a given system Volume at a certain Temperature) are performed on all models. In this method, the potential of fluid-fluid is first obtained; the equations of motion of atoms are then solved based on the following Newton’s equations:

mi

d 2 ri dt 2

Fi

' i ĭ .

(2)

The potential for the given system Ɏ is assumed to be the sum of the effective pair potentials I (rij ) from Eq. (1) as ĭ

¦¦ I (r ) , i

j !i

ij

(3)

where rij is the distance between atoms i and j. Before starting the simulation, initial positions and velocities are assigned to all particles in the system. Then at each step intermolecular potentials and forces are calculated. This is the most time-consuming step in typical MD simulations. The equations of motion are integrated using time integration algorithms that are based on finite difference methods. The Verlet integration rule is applied as G n 1 r

G n G n 1 G 2r  r  't 2 a (t )  2('t 4 ) .

(4)

Once the equations of motion are integrated, the relevant properties of the system are calculated and stored[16]. In the present study, two identical droplets, each containing 864 L-J molecules, are formed and are placed in a cubic space with volume of 40×40×40 dimensionless length units under a temperature of 90 K. Initially the space is a vacuum, the radius is about 6 units length, and the L-J potential is given. The parameter values of argon in Eq. (1) are given as mL = 6.63×10í26 kg, ıL = 0.3405 nm, İL = 1.67×10í21 J, and c = d = 1, respectively. The argon molecules are distributed initially on the lattice sites in the cells of fcc (100). Periodic boundary conditions are applied in three coordinate directions for a 3D system, where there exist two liquid droplets and their vapour. Detailed structure can be

Yan and Ji: Molecular Dynamics Simulation of Behaviours of Non-Polar Droplets Merging and Interactions with Hydrophobic Surfaces

found in books on Solid State Physics (e.g. Kittel, 2005[17]). The length unit is of ıAr= 0.3405 nm. The system of the present simulation is evenly divided into 50 slabs. The dimensionless density of molecular numbers is defined as U * NV L3 / V , where N represents the number of molecules in each slab, V is the volume of each slab. 2.2 Wetting phenomena A system with L-J fluid confined between two solid walls is studied using non-equilibrium molecular dynamics (NEMD) simulation method[16]. Fig. 1 shows a snapshot of the system at steady state. The size of the cell is of 5.83 nm u 3.85 nm u 7.22 nm, including both the two solid walls and the fluid confined between them. The distance between the two walls (in z direction excluding the solid walls thickness) is 5.41 nm. The focus in the present study is to investigate the wetting phenomena of the fluid-solid near the lower wall, which occur within a very small length scale. The current vertical distance can provide a sufficiently large space for the molecules evaporating away from the heating wall and condensing near the cooling wall. Periodic boundary conditions are applied in x and y directions. All the variables including density, potential energy are computed in the slices of x-y planes along z direction; so a smaller length scale in y direction is used to save computing time and storage. For both the fluid and the solid molecules, the L-J potential function is used to calculate the intermolecular forces. This is an appropriate first step to understand the

Fig. 1 Snapshot of NEMD simulation system.

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phenomenological influence of realistic atomic granularity on the liquid-vapour-solid system and suchlike. Therefore, for the sake of physical understanding, in the present study, argon is used as the L-J fluid with the following potential parameters: mL= 6.63×10í26 kg, ıL = 0.3405 nm, İL=1.67×10í21 J. The solid wall is represented by four layers of face centred cubic (fcc (111), see the inset of Fig. 1) surface of harmonic platinum molecules with parameters as: mS = 3.24×10í25 kg, ıS = 0.2475 nm, İS = 8.35×10í20 J. This type of crystalline structure is quite stable so it is not necessary to introduce any additional phantom molecules with springs holding the solid molecules at the crystalline sites. When dealing with wetting in MD simulations, the adjustable parameters c and d in Eq. (1) are assigned to different values to correspond to different wetting ability of solid-liquid interaction. For solid-liquid interaction, the potential is in the form as

ISL (r ) 4H SL ^cSL (V SL / r )12  dSL (V SL / r )6 ` .

(5)

In a previous study, Ji and Yan[16] applied H SL H SH L and V SL (V S  V L ) / 2 to a completely wetting (hydrophilic) system when both c and d are maintained as unity. In the present study, we set c = 0.1, d = 0.05, in line with a hydrophobic system.

3 Results and analysis As a first step, two identical droplets are initially put into a vacuum cubic where the droplets evaporate to fill the vacuum space and interact with each other under different distances. Interactions of fluid with hydrophobic walls are studied as well to reveal the liquid-solid interactions from the microscopic viewpoints. 3.1 Droplet merging When the distance between two identical droplets is relatively short such as 12 dimensionless length units between the two mass centres, the merging process is shown in Fig. 2. Within a very short period of time, the two droplets touch each other and at the time of 5×10í12 s, the two become connected because of attractive interactions from the molecules in each cluster (Fig. 2a). A

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dumbbell shape is formed at the time of 1.5×10í11 s, while some molecules dissipate to the vacuum space of the simulation system (Fig. 2b). The interfacial shape connecting the two droplets then becomes smooth so that the merged droplets appear to be elliptic and more molecules occupy other spaces of the system as vapour (Fig. 2c). In Fig. 2d, with the continuous effect of intermolecular attractive forces, the elliptic droplet adjusts its shape: the major axis of the ellipse decreases while the minor axis increases. Eventually the major axis and minor axis are almost the same in length (Fig. 2e). Fig. 2f shows a merged larger liquid droplet in equilibrium with its vapour phase in the simulation cubic system. If the initial distance between the two droplets is a little larger such as 14 dimensionless length units be-

(a) t = 5×10í12 s

(c) t = 2.5×10í11 s

(e) t = 1×10í10 s

tween the two mass centres, the droplets will not touch each other as fast as shown in Fig. 2. At 5×10í12 s, some molecules in each cluster move close towards each other while others remain a distance between two droplets (Fig. 3a). While some molecules are dissipating to other positions in the cubic system, the two droplets adjust their respective shapes to be spherical (Fig. 3b). When the molecules near the boundary of one liquid body thermally fluctuate into the range of attraction of the other droplet, the merging occurs by forming a string of mutually attracting molecules (Fig. 3c). These molecules are gradually thickened into the connecting part of a “dumbbell” (Fig. 3d). The two ends of the “dumbbell” are then smoothly combined in a zipper-like merger to make an elliptic merged droplet (Fig. 3e). In the last time stage a spherical larger droplet is formed (Fig. 3f).

(b) t = 1.5×10í11 s

(d) t = 5×10í11 s

(f) t = 3×10í10 s

Fig. 2 Process of droplets merging (short distance).

(a) t = 5×10í12 s

(b) t =1.5×10í11 s

(c) t = 2.5×10í11 s

(d) t = 5×10í11 s

(e) t = 1×10í10 s

(f) t = 3×10í10 s

Fig. 3 Process of droplets merging (large distance).

Yan and Ji: Molecular Dynamics Simulation of Behaviours of Non-Polar Droplets Merging and Interactions with Hydrophobic Surfaces

For a larger distance (16 dimensionless length units between the two mass centres), the droplets will not touch each other easily as shown previously in Figs. 2 and 3. On the other hand, the two droplets adjust their respective shape in their own position while some molecules dissipate into the vacuum of the system (Fig. 4). Because there is no external force or initial relative velocity, the molecules in each cluster cannot escape the attractions from the molecules in their own clusters. Two individual droplets adjust their own shapes and move independently in the simulation system, with the coexistence of their vapour phase. Simulations for different system temperatures other than 90 K are also carried out. No obvious effects for the temperature on the merging characteristics (mainly the critical distance where merging can occur) are found.

(a) t = 5×10í12 s

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The instantaneous density of molecular numbers of the droplets and their vapour in the system are calculated based on the molecules’ distribution. As shown in Fig. 5, the change of the dimensionless density of molecular numbers along x direction of the system is obtained. Due to an instantaneous value rather than the statistical average one, the density distributions display a kind of fluctuation along x direction. Nevertheless, the liquid-vapour interface can easily be distinguished from the bulk liquid and vapour as shown in Fig. 5 as an example, which provides a quantitative description for the qualitative illustration of the results of droplets merging (with short distance) evolutions shown previously in Fig. 2. In the below diagrams, the density distribution along the droplet interface in x direction is clearly demonstrated. Each diagram in Fig. 5 is exactly inline with the droplet evolution shown in Fig. 2. It is shown that the density reaches to the peak when the merging process is completed.

(b) t = 1.5×10í11 s

(a) t = 5×10í12 s

(c) t = 2.5×10í11 s

(e) t = 1×10í10 s

(d) t = 5×10í11 s

(f) t = 3×10í10 s

Fig. 4 Two droplets without merging.

(b) t = 1.5×10í11 s

Fig. 5 Density distribution of the droplets and their vapour in the system (short distance).

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(c) t = 2.5×10í11 s

(d) t = 5×10í11 s

(e) t = 1×10í10 s

(f) t = 3×10í10 s

Fig. 5 Continued.

3.2 Wetting phenomena Previously the wetting phenomena were studied only by taking into account surface contact angle and simulated by LBM[3,4]. In the present study, the simulation system of fluid is confined between walls, fluid-solid interactions can therefore well be explored. Different values can be selected and adjusted to correspond to different strength or weakness of these interactions, which will be as a reasonable simulation model to be in line with the real practical cases for different materials and surface properties. In the present study, we choose an ideal very hydrophobic wall as an initial investigation to show the microscopic view by MD simulation methods. We compare the wetting characteristics of hydrophobic wall with that of hydrophilic wall, the latter has been fully studied in the authors’ earlier work[16]. For better understanding, the effect of heating wall is also investigated. Fig. 6 shows the snapshots of hydrophobic systems under different temperature conditions. The focus in the present study is on the region of fluid near the lower wall, which is highlighted by an elliptic circle in Fig. 6. When the solid wall is hydrophobic, there will not form a liquid film in the vicinity of the wall, instead the liquid molecules move away from the lower wall because they have large kinetic energy to escape the attraction of the solid wall (Fig. 6a). Therefore, the barrier which limits the hydrophilic molecules is able to be overcome. Furthermore, when the lower wall is heated, fewer molecules are left in the region near the lower wall (Fig. 6b), rather than the persistent accumulated liquid film above the heating hydrophilic wall as shown in Ref. [16].

(a) 100K-100K system

(b) 110K-90K system

Fig. 6 Snapshots for hydrophobic systems under different temperature conditions.

Yan and Ji: Molecular Dynamics Simulation of Behaviours of Non-Polar Droplets Merging and Interactions with Hydrophobic Surfaces

As a comparison, Figs. 7 and 8 provide the density distributions of hydrophobic and hydrophilic systems in the same temperature conditions. When the upper and lower walls are maintained both in 100 K, the density distributions show symmetry in vertical direction. For the hydrophilic system, the density of the fluid in the vicinity of solid wall is significantly larger than those in the vertical centre of the system (Fig. 7a). For the hydrophobic system, the density of the fluid near the solid wall drops dramatically compared to the density in the same position of hydrophilic system (Fig. 7b). When the lower wall is heated to a temperature of 110 K, the liquid film with high density near the lower wall becomes thinner but the density there remains at a high level for the hydrophilic system (Fig. 8a); whereas for the hydrophobic system, the density of fluid near the lower wall reduces sharply to leave only a very slight peak compared to the bulk density (Fig. 8b). It is shown from the above results that interactions of fluid and hydrophobic walls exhibit a drastic difference from hydrophilic walls. Detailed images are

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revealed by MD methods which are almost impossible for the conventional CFD method or mesoscopic methods such as LBM. MD method is therefore a promising way for detail investigation, particularly when its advantages within length scale and time scale can be utilized to its utmost in a proper and reasonable way.

(a) Hydrophilic

(b) Hydrophobic (a) Hydrophilic

Fig. 8 Density distribution for 110K-90K system under different wetting conditions.

4 Conclusion remarks

(b) Hydrophobic

Fig. 7 Density distribution for 100K-100K system under different wetting conditions.

MD simulations have become an efficient tool of microscopic insight to reveal the physical mechanisms of the transport phenomena of multiphase flow in microchannels. The merging of two droplets apart from each other at different initial distances is tested numerically and the possible larger (or critical) dimensionless distance up to 14 units for the droplets merging is discussed. The

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evolution of the merging process is simulated numerically and presented in the paper. In the present modelling, the molecules near the boundary of one liquid droplet thermally fluctuate into the range of attraction of the other droplet, forming a bridge to connect the two droplets. A dumbbell shape is then formed and thereafter an elliptic merged droplet. Eventually a larger merged spherical droplet appears in the system and is in equilibrium with its vapour phase. A more realistic simulation system needs to be established on the basis of the current preliminary results for simulating mist flow of microchannel boiling. To do so, background media as vapour need to be assigned into the system to replace the preliminary vacuum system. Different initial droplet size (particularly large cluster with more molecules) and shape, external forces, initial assigned relative velocity, etc. are the major factors to be considered further. Moreover, two different droplets (not identical), three or more droplets can be simulated to investigate the characteristics of their merging process. In addition, the system with the effect of solid wall on droplet merging, and with more realistic molecules (water or other multi-atomic molecules) should also be studied. Some of the works are undergoing and will be reported later. Interactions of fluid and hydrophobic walls are studied and compared with those of hydrophilic walls. Density distributions near the wall become significantly smaller in fluctuation in the case of hydrophobic walls for the isothermal system. With the heating of the lower wall, the density of fluid near the lower hydrophobic wall decreases to exhibit a very slight peak, in comparison with the large fluctuations of fluid density near the hydrophilic walls.

Acknowledgement The work was supported by the UK EPSRC under grant EP/D500125/01.

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