Molecular Dynamics Study of Thermally Augmented Nanodroplet

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Molecular Dynamics Study of Thermally Augmented Nano-Droplet Motion on Chemical Energy Induced Wettability Gradient Surfaces Monojit Chakraborty, Anamika Chowdhury, Richa Bhusan, and Sunando DasGupta Langmuir, Just Accepted Manuscript • Publication Date (Web): 18 Sep 2015 Downloaded from http://pubs.acs.org on September 19, 2015

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Molecular Dynamics Study of Thermally Augmented Nano-Droplet Motion on Chemical Energy Induced Wettability Gradient Surfaces Monojit Chakraborty, Anamika Chowdhury, Richa Bhusan, Sunando DasGupta* Department of Chemical Engineering, Indian Institute of Technology Kharagpur 721302.

*corresponding author Email

[email protected]; [email protected]

Ph:

+91 - 3222 - 283922

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Abstract Droplet motion on a surface with chemical energy induced wettability gradient has been simulated using molecular dynamics (MD) simulation to highlight the underlying physics of molecular movement near the solid-liquid interface including the contact line friction. The simulations mimic experiments in a comprehensive manner wherein micro-sized droplets are propelled by surface wettability gradient against forces opposed to motion . The liquidwall Lennard-Jones interaction parameter and the substrate temperature are varied to explore their effects on the three-phase contact line friction coefficient. The contact line friction is observed to be a strong function of temperature at atomistic scales, confirming their experimentally observed inverse functionality. Additionally, the MD simulation results are successfully compared with those from an analytical model for self-propelled droplet motion on gradient surfaces.


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1. Introduction Self-propelled droplet motion on gradient surfaces for specialized applications has emerged as an area of active research in the field of microfluidics and nano-fluidics with the potential to transport droplets; eliminating the need for pumps1 and other external devices2. Apart from the broad domain of microfluidic devices, mobile droplets are relevant in many natural phenomena as well as new and emerging applications3,4 Some of the living organisms in low-rainfall areas have naturally occurring wettability gradient surfaces to collect water from the humid atmosphere.5 Similar super-hydrophobic surfaces are now being used extensively in the textile industry to develop self-cleaning and water repellent surfaces.6 Bio-inspired materials with characteristic wettability have applications ranging from anticorrosion, antifogging, liquid transport, biomedical etc.7 Surface energy gradient surfaces are used to investigate the behavior and kinetics of protein adsorption/desorption and are relevant in cell attachment, growth and proliferation studies.8,9 In condensation heat transfer processes, continuous movement of condensed water droplets away from the cold surface is achieved using wettability gradient surfaces to maintain high heat transfer coefficient and even in micro-gravity situations.10 Brochard11 proposed a theoretical expression for the velocity of droplet motion on surfaces with wettability gradient, whereas the experimental observations were first reported by Chaudhury and Whitesides12. They concluded that the motion was a result of the surface tension difference between the two edges of the droplet on wettability gradient surfaces. Working on similar concepts, Cao et al. have shown that it is possible to create a continuous and spontaneous antigravity water delivery system.1 Daniel and Chaudhury also demonstrated that droplet movement over wettability gradient surface may be augmented by 3 ACS Paragon Plus Environment

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subjecting it to mechanical vibrations in the form of square waves.13 Wang et al. showed a surface vibration triggered directional transport of droplet on wettability gradient surface, even at low temperature and high humidity.14 The thermal gradient induced movement of droplets on hydrophobic surfaces was demonstrated by Brzoska et al.15. Chakraborty et al. in their recent experimental study have shown that the velocity of the self-propelled droplets on a gradient surface can be substantially enhanced at elevated temperatures.16 There have been several other studies for actuating droplet motion using optical,17 acoustic,18 electrical energy,19 and morphology gradient surfaces.20,21,22 Ford and Nadim23 used a theoretical approach to report droplet velocity profile on a substrate by the application of thermal gradient for drops with small Peclet numbers. In 2008, Halverson et al.24 used molecular dynamic (MD) simulations to report the steady movement of droplets on uniform wettability gradient whereas, in the case of non-uniform gradients, the droplets were observed to be pinned. Inclusion of contact angle hysteresis led to better agreement of droplet velocities with the theoretical predictions. In the present study, the dynamics of nanodroplet movement over gradient surfaces is obtained from first principles. The liquid-substrate Lennard-Jones interaction parameter, εls, and the substrate temperature have been varied to explore their effects on the three-phase contact line force, opposing droplet motion. Changes in εls and substrate temperature cause

significant alterations in the movement of the molecules and is expressed through the

changes in the net molecular displacement frequency, к0. Such variations in к0 lead to a significant change in the three-phase contact line friction co-efficient (ζ). Furthermore, the MD simulation results are successfully compared with an analytical model specifically developed to describe self-propelled droplet motion on gradient surfaces.


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2. Theory Molecular kinetic theory examines the dynamics of molecules near the three-phase contact line. The driving force for the contact line motion arises due to the imbalance of the surface tension forces on the wettability gradient surfaces, which disturbs the adsorption equilibrium. Suda and Yamada25 measured this driving force using a flexible glass microneedle and concluded that the hydrodynamic force is not sufficient to balance the driving force. Instead, they proposed an alternative friction theory, wherein the liquid near the substrate may be considered as an assembly of N small solid-like domains. The friction between these domains is said to counteract the driving force. Thus the forces that oppose the motion are primarily the hydrodynamic force and the three-phase contact line friction force with later being the dominant one.16 A detailed description of all the forces are given in section 4.2. The three-phase contact line frictional force is molecular in nature and it arises due to the pinning of the molecules at the three-phase contact line.26 The origin of this resistive force is due to the dissipation of the internal energy of the molecules near the contact line. The liquid molecules adsorbed on the solid substrate, along the three-phase contact line, jump in the forward and backward directions with frequencies к+ and к− respectively and the net displacement frequency26 is the resultant of the two. At equilibrium, both к+ and к− are equal and are denoted by к0. The velocity of the contact line is given by the net displacement frequency multiplied by λ, where λ is the average length of one molecular displacement. In the case of moving contact line, work must be done to overcome the energy barrier of the molecular movement in a specific direction. The driving force for the molecular displacement is a result of the out of balance surface tension force, equal to Fw= γ (cosθS ‒ cosθD) at the non-equilibrated contact line, where θS and θD refer to the static 5 ACS Paragon Plus Environment

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and dynamic contact angles. Based on these ideas, the expression for the forward velocity of the contact line has been proposed as26,27

 γ (cos θ S − cos θ D )λ 2  U = 2k 0λ sinh   2 k BT  

(1)

Where γ is the liquid-air interfacial tension, к0 is the equilibrium frequency of the random molecular displacements and a function of the activation free energy of wetting26. If the argument of the hyperbolic sine is very small, the equation reduces to

k 0γ (cos θ S − cos θ D )λ 3 U= k BT

(2)

where к0, λ, kB and T are clubbed together in a single parameter, denoted by ζ and is termed as the three-phase contact line friction coefficient as,27

ζ =

k BT k 0λ 3

(3)

It has been extensively reported in the literature that ζ can only be determined in situ, using experimental data, and depends strongly on the temperature and κ0. In the present work, molecular dynamics is used to simulate the dynamics of droplet motion on a chemical energy induced wettability gradient surfaces. Furthermore, the effects of temperature and wettability on ζ and hence the droplet motion are probed as well.

3. Simulation Details The simulations reported herein were carried out using GROMACS-4.0.728,29,30 whereas Visual Molecular Dynamics (VMD)31 was used for the intermittent visualization of the droplet configuration. The well-known Lennard-Jones [Eij(r)] potential was used to model the system as,


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 r ij Eij (= r ) 4ε ij    σ ij 

  

−12

 rij − σ  ij

  

−6

 qq +C i j rij  

(4)

Where ε = depth of the potential well; σ = finite distance at which the inter-atomic potential is zero; r = inter-atomic distance ; C= coefficient proportional to the electrostatic constant and qi ,qj= charge on ith and jth atom. The i and j subscripts are used for specifying the component species and can be replaced by the corresponding atoms at relevant places. The Simple Point Charge (SPC) model32 was employed for the simulations, wherein the water molecules were assumed to be of ideal tetrahedral shape with a H-O-H bond angle of 109.470. The oxygen molecules and the oxygen-wall interactions were assumed to be of L-J type, whereas the interactions between the solid atoms were excluded. Majority of the water models used for the molecular dynamic study of water substrate interaction have εOH and εHH set to be equal to zero33 including the model used herein (SPC). As a consequence, the calculated value of εSiH turned out to be zero as well, using the Lorentz-Berthelot mixing

ε SiH rule (=

ε SiSi × ε HH ).34,35 The other parameters used were σoo=0.3166 nm36 and for Si

atoms, σSiSi=0.3408 nm37According to the Lorentz-Brethelot mixing rule34, the value of σls can be calculated as

+ l  ss s ls =  s   2 

(5)

Thereby for the reported simulations, the value of σls (here subscript l stands for the oxygen atom of the liquid phase and s stands for the Si atom of the solid phase) was fixed at 0.3276 nm. A cut-off length of 1.2 nm35 was used for the Coulombic and the van der Waals forces. Periodic boundary conditions were applied in both X and Y directions. No changes in the simulation results were observed by changing the cut-off length to 1.1 nm, 1.5 nm and 1.7 7 ACS Paragon Plus Environment

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nm. Therefore the water droplet maintained a minimum distance greater than 1.7 nm from the edges of the substrate throughout the simulation whereas the cut-off lengths for both coulombic and vander-waal forces were maintained at 1.2 nm. This conmirms the validity of periodic the boundary condition for the system used herein. The temperature of the system was maintained using the v-rescale coupling algorithm38 with a time-constant of 0.1ps for the water droplet. The force field applied was 53A6 of GROMOS-96 version39 and the steepest descent type integrator was used. The simulations for the movement of water molecules on the gradient surfaces were performed for a time scale of 3000ps and the spreading dynamics of the droplet on a uniform wettability surfaces was observed for 2000ps. The dimensions of the simulation box used were 35 nm×35nm×10 nm, comprising of around 12000 water molecules and around 15000 Si atoms, which were found to be optimum, based on computation time and simulation results. The simulations were performed using NVT (at different constant temperatures) ensemble40,35. 3.1 Spatial variation of Epsilon- The wettability gradient has been created solely by spatial variation of chemical energy on the surface. The magnitude of the interaction parameter (εls) is an indicator of the attractive force between the solid and the liquid atoms and its strength is directly proportional to the substrate hydrophilicity. To study the variation of the contact angle with epsilon, a water drop at equilibrium on a homogenous surface was simulated for different values of epsilon (εls). The equilibrium contact angles were calculated from the extracted images of the simulated water droplet using VMD and ImageJ ( version 1.41 ) software. The contact radius of the equilibrated drop was computed using coordinates obtained from the simulation. The variation in the contact angle and contact radius as a function of the interaction parameter is shown in Figure 1.


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Figure 1 Variation of contact angle and contact radius as a function of epsilon The observed reduction in contact angle with increase in εls corroborates the fact that substrate hydrophobicity decreases with increase in the attractive forces between the substrate and the water droplet. The other component of the interaction potential (σls) cannot be altered since the minimum cut-off distance specification imposes a constraint on the inter-particle distance. The interactions among the wall molecules were excluded while performing the simulation studies. 3.2 Creation of the Gradient Surface- A substrate (34 nm × 34 nm) was created consisting of two layers of Si atoms in cubic crystal structure in the x-y plane. Parallel strips (ydirection) were used to mimic the wettability gradient (x direction) surface. The values of 9 ACS Paragon Plus Environment

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the interaction parameter (εls) for these parallel strips were varied (See Supplementary information 1) to create a surface with gradients in wettability (Table 1). The choice of a specific wettability gradient was made to mimic the experimentally observed spatial variation of the driving force, where it was observed that the driving force increased initially, reached a maxima followed by a decrease.16,41 In the current molecular dynamic simulations, the spatial variation of the interaction parameter (εls) is chosen to mimic (explained in detail in subsection 4.1.1) the resultant variation of the contact angle differences similar to those encountered in the experiments.12,16,41 The strip dimension in the y-direction was chosen to ensure the spreading and motion of the droplet in one (x) direction only. The simulation runs were performed for the water molecules, keeping the wall atoms frozen at their positions. Table 1: ε values as a function of x-coordinate to create a wettability gradient surface


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4. Results and Discussion The results section is divided into four subsections. In the first subsection (4.1), the movement of the droplets on a gradient surface is explored using MD simulations and a spatial variation of the instantaneous droplet velocity is determined. The effects of variations in the liquid-solid interaction parameter (εls) and substrate temperature on the droplet velocity are examined as well. In the second subsection, an analytical approach is used to characterize the movement of the droplets on gradient surfaces at different temperatures. Subsection 4.3 deals with the evaluation of the contact line friction coefficients, ζ, by analyzing the symmetric spreading of a sessile droplet. A comparison of ζ, obtained from the analytical model with that from the MD simulations is presented in the last subsection. 4.1 Simulation of Droplet Motion on Gradient Substrate The simulations of droplet motion over the created gradient surface are performed with a run time of 3 ns. A cuboidal block of water molecules is initially positioned towards the hydrophobic end of the substrate while the temperature is kept constant at 300K. The energy of the system is first minimized allowing the droplet to settle on the gradient and only then it is allowed to equilibrate. The droplet starts to move with distinctly different values of contact angles at the front and the rear end. Figure 2 represents the droplet motion from the hydrophobic end towards the hydrophilic end on the substrate.


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Figure 2 Simulation of droplet motion at 300K. The mean density of the simulated droplet remains close to the actual density of water (see Supplementary Information 2) throughout the simulation run. It was shown that the droplet velocity may also be affected by several other factors such as the presence of surface roughness induced wettability gradient20 and the impact velocity of the droplet on a substrate.42,43 However, for the molecular approach used in the present study, the dimensions of the droplet are in nanometers, comparable to the dimensions of the surface roughness generally encountered experimentally including the studies referenced above. Herein, the only cause of the droplet movement on a smooth surface is an imposed chemical energy gradient, induced by the alteration of the solid-liquid inter-atomic interaction parameters. 4.1.1. Effect of the interaction parameter (εls) on the velocity profile of the droplet: 12 ACS Paragon Plus Environment

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The driving force for the droplet motion is directly proportional to the difference in the cosine of the contact angles at the advancing and receding fronts.16,41 With higher values of ∆ε across the advancing and receding ends, the droplet experiences larger difference in the contact angles and higher values of the driving force. Using this concept, three different substrates with varying hydrophobicity gradients (∆ε/∆x) have been designed for the simulations. The three substrates are different in terms of the values of the wettability gradient, ∆ε/∆x, along the direction of the droplet motion (x-direction).The gradient is maximum for substrate 3 and minimum for substrate 1. The spatial variations of εls and ∆εls are shown in Table 1, whereas the variations in wettability gradient (∆ε/∆x) is depicted in Figure 3.

Figure 3 Variation of ∆ε gradient in the direction of the droplet movement for three different test substrates. The lines are guide to the readers’ eyes only. The co-ordinates of the center of mass along the direction of the movement of droplet, on three different wettability gradient substrates are determined at every time instant from the 13 ACS Paragon Plus Environment

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output trajectory file, where the center of mass is defined as that coordinate which minimizes the mass weighted sum of a periodic spatial function such as a cosine function for the group of atoms which are eventually a part of a larger system.44 The displacements of the center of mass along the direction of wettability gradient are evaluated (see Supplementary Information 3). It is observed that the displacements along other directions are negligible (see Supplementary Information 4 and 5). The instantaneous velocities of the droplet are obtained from the displacement-time curve. Since the driving force for droplet transport is related in a non-linear fashion to the wettability gradient, the apparent minor changes in the values of the gradient in Figure 3 may result in appreciable changes in the driving force and hence the velocity. For example, a ~7.8% increase in the peak value of the gradient (∆ε/∆x) between substrate 1 and substrate 3 leads to a ~29% increase in the peak value of the droplet instantaneous velocity (Figure 4). It is clear from Figures 3 and 4 that the droplet motion can be controlled by tuning the ∆εls gradient.

However, for all subsequent simulations of droplet motion including that at

elevated temperatures (next subsection onwards), the substrate with the highest wettability gradient (Substrate 3) is used.


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Figure 4 Velocity Profiles for three different substrates at 300K; Simulation data at every 20th point are shown for clarity. The velocity profile, obtained from the simulation results, initially shows an increasing trend till it reaches a maximum and then starts to decrease. The characteristics and trends in velocities obtained from these simulations are remarkably similar to the experimental trends and are in accordance with the displacement-time curve inflection point.16,41 The velocities obtained in the simulations are of the order of 3-4 m/s, which is an order of magnitude higher as compared to the velocities reported in the experiments.16 It has been reported earlier that decreasing the size of the droplet from millimeter to nanometer scale can lead to significant increases in the droplet speed (up to three orders of magnitude)24 and is attributed to the reduction in the inertial resistance. 4.1.2 Effect of temperature change Simulations of droplet movement at different temperatures (in steps of 2K from 298 K to 310 K) are performed on substrate 3 (as specified in Figures 3 and 4) to study the effect of 15 ACS Paragon Plus Environment

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the temperature change. The simulation results are presented in Figure 5 and Table 2, showing a distinct increasing trend of droplet movement with temperature.

Figure 5 Variation in velocity with temperature on substrate 3; simulation data at every 20th point are shown for clarity. To enhance readability, date at only three temperatures are presented here. Table 2 Variation of the peak velocity with temperature on substrate 3 Temperature (K)

Peak Instantaneous Velocity (m/s)

298

3.896

300

4.003

302

4.118

304

4.450

306

4.580

308

4.712

310

4.913


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The temporal variations of the advancing and receding contact angles along the path of the moving droplet are measured from the extracted images of the of the droplet movement using ImageJ (version 1.41) at 298K and are presented in Figure 6. The differences in the contact angles grow rapidly with time resulting in the initiation of the droplet movement The increase in the difference between the cosines of the contact angles at the two ends (as can be seen in the inset of Figure 6) provides the driving force for droplet movement and correlates well with the droplet velocities (Figure 5) mimicking the experimentally observed trends.16,41 The increase in velocity with temperature is attributed to the decrease in friction associated with the contact line motion. The contact line friction coefficient (ζ) is used to quantify the resistive forces due to friction and is expressed as a combination of several parameters.16,26,27 The values of the coefficient are evaluated, as a function of temperature, from the predicted theoretical model.

Figure 6. Spatial variation of contact angles at 298K


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4.2 Theoretical model The instantaneous velocity of the droplet on a wettability gradient surface can be evaluated by analyzing the surface tension gradient inspired driving force and the resistive forces acting on the drop, namely the hydrodynamic and the three phase contact line forces. A force balance on the moving droplet results in the following equation,

m

dU = Fdriving − [ Fh + Fcl ] resistive dt

(6)

where, m is the mass of the moving droplet (taken as the total mass of 12000 water molecule) ; Fh = hydrodynamic force; Fcl = three-phase contact line friction force; The droplet also experiences a drag force if it is encapsulated by a filler medium. However, this drag force is found to be insignificant when compared to the other resistive forces and is neglected.16 The driving force acting on the droplet is given as 16,41,45,

Fdriving = 2 R(t )g g (θ ) π /2

where = g (θ )

∫ [cos (θ )

a

(7)

− cos (θ )r ] cosφ dφ

0

R(t) is the radius of the footprint of the drop and γ is the liquid-gas interfacial tension. Subramanian et al.45 used a wedge approximation to the Stokes flow condition analysis by Cox46 to predict the hydrodynamic force exerted by the solid surface on the liquid drop as,

Fh (8µ R (t ) f (θ , ∋)U ) =

(8)

where, (1−∋ )

f (θ , ∋) = ∫ 0

1 (1 − Y 2 ) tan 2 θ [ ln((1 − Y 2 ) − ln ∋] 2 dY 2 2 2 2 −1 [tan θ (1 − Y ) − (1 + {1 − Y }tan θ ) tan (tan θ (1 − Y )] 18 ACS Paragon Plus Environment

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U(x) is the instantaneous velocity of the drop; θ is the dynamic contact angle of the drop and ϶ =Ls/R where, Ls is the slip length41. MD simulations depicting similar physical situations as in the present study indicated that the slip length values are of the order of 0.5 nm41. As the droplet moves along the path of decreasing hydrophobicity, dynamic wetting can be visualized as the displacement of the three phase contact line from one location to another. The advancement of the three phase contact line leads to dissipation of energy at the molecular level26 and is commonly known as the three phase contact line friction force (Fcl), and is expressed as47,

Fcl = 2 P ( t ) ζ U

(9)

Where, P(t) is the droplet perimeter length and ζ is the coefficient of contact line friction16,26,27 Substituting equations (7), (8) and (9) in equation (6), a variation of droplet velocity U is obtained as,

dU = m 2 R (t ) g g (θ ) − 8m R(t ) f (θ , ∋ ) + 4π R ( t ) ζ  U dt

( 10 )

Integration of equation (10) with the initial condition U (t=0) = 0 yields,

U

 8m R(t ) f (θ , ∋ ) + 4p R ( t ) ζ  × t    × 1 − exp  −     m 8m R(t ) f (θ , ∋ ) + 4p R ( t ) ζ      (11) 2 R (t ) g g (θ )

As the variation of the position of the center of mass of the droplet with time is known, the spatial variation of the droplet velocity can also be predicted from equation (11). The variation of all the parameters with time is known from the simulation results, except ζ. For a given solid–liquid system, there is no definitive way of predicting the values of ζ 19 ACS Paragon Plus Environment

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(responsible for the dynamic wetting behavior) from independently measured quantities27,48 and must be evaluated in situ for each simulation. The values of ζ as reported in the literature vary considerably, from 0.000089 Pa.s for molecular dynamics simulations of dynamic wetting behavior of liquid to 738000 Pa.s for melting of metal and oxides on Molybdenum.48,49,50,51 Herein, the values of ζ are evaluated by minimizing the root mean square error between the theoretically predicted velocities (equation 11) with the MD simulation (Figure 5) results. Reasonable agreement between the predicted velocities (using equation 11) with the velocities obtained from MD simulations is observed and presented in Figure 7.

Figure 7 Comparison of velocities between the simulation and theoretical studies (Droplet velocities are calculated using equation 11. Data at every 20th point are shown for clarity. To enhance readability, comparisons at only two temperatures are presented here. A decrease in ζ with temperature is observed which is in tune with the experimentally observed trend.16 For the simulations reported herein, the values of ζ are found to be in the


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range of 0.0039 ─ 0.0052 Pa.s, comparable with those reported by other researchers48-51 from similar studies. To validate the variation of ζ with temperature a detailed analysis of droplet spreading on uniform wettability surfaces is performed and presented in the next section. 4.3 Spreading of droplet on substrates with uniform wettability The three-phase contact line friction coefficient, ζ is calculated, from its fundamental definition (using equation 3), for a series of simulation of droplet spreading on substrates having spatially invariant wettabilities. The effect of temperature and surface wettability on the contact line friction coefficient are evaluated keeping the other parameters constant. The procedure adopted for the evaluation of ζ is similar to that used by Bertrand et al.49 In terms of molecular displacement, the advancement of the contact line can be represented as molecules jumping from one site to another with a frequency κ0 and the average length of one molecular displacement distance, λ as discussed earlier. Near the wall, the molecular jumps are anisotropic in nature, while far away the jumps are equally probable in all directions. Thus, jump frequencies can be primarily divided into two category49; the parallel frequency (𝜅∥ ) and the perpendicular frequency (𝜅⊥ ).

4.3.1 Density Profiles: 𝜅⊥ and 𝜅∥ are characteristic of the first two layers of the water

droplet. The number density profile in a direction normal to the wall surface has been plotted 52,49 (Figure 8), at a constant temperature of 298K to identify the first two layers. The density profile of the equilibrated drop has been evaluated at a time of 1600 ps since no significant temporal change in the droplet profile is observed thereafter. The observed reduction in the droplet apex height with the increase in hydrophilicity (inset of Figure 8) indicates an enhancement in the attractive forces between the wall Si atoms and water


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molecules. Close to the surface of the wall, the droplet demonstrates distinctly high number density.49 However, the number density becomes equal to the bulk number density after about 10Å (inset of Figure 8). The first layer density remains nearly constant with variation in temperature (at constant ε).

Figure 8 Density variation with distance from the wall surface plotted at 298K. To enhance readability, variation for only three ε are presented here

4.3.2 Parallel frequency (𝜿∥ ): To evaluate the parallel frequency, the molecular jumps are defined as the displacement of a water molecule of the first layer by a distance not less than

𝜆∥ in planes parallel to the wall surface. The jumps near the wall is attributed to adsorption on the wall surface.49 𝜆∥ is approximated as the largest distance between two consecutive

wall atoms (adsorption sites),49 which is 4√2Å for the system used herein. The cumulative

percentage of molecules undergoing the first parallel jump, as a function of time, is plotted 22 ACS Paragon Plus Environment

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(Figures 9 and 10) for different combinations of surface hydrophilicity (εls) and temperature. The time when 50% of the molecules get displaced by at least 𝜆∥ is halved to take into account the possibility of both forward and backward displacements49 and is then inversed to find the required frequency 𝜅∥ .

Figure 9 Cumulative percentages of water molecules displaced by λ as a function of time at 298K, while varying interaction parameter, ε, values. (A), (B), (C) are cumulative percentage parallel displacement for ε = 4.02, 6.54 and 8.66 respectively and (D), (E) and (F) are cumulative percentage perpendicular displacement for ε = 4.02, 6.54 and 8.66. Only three values of ε are shown, to enhance readability.


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Figure 10 Cumulative percentages of water molecules displaced by λ as a function of time while varying temperature at constant surface hydrophilicity (ε = 6.54; Contact angle 7.67°). (A), (B), (C) are cumulative percentage parallel displacement at 310 K, 304 K and 298 K respectively and (D), (E) and (F) are cumulative percentage perpendicular displacement at 310 K, 304 K and 298 K Only three values of temperature are shown, to enhance readability 4.3.3 Perpendicular frequency (𝜿⊥ ) : A perpendicular jump is defined as the displacement

greater than 𝜆⊥ in the direction perpendicular to the wall plane.49 The distance between the first two crests in the number density plot (Figure 8) is quantified as 𝜆⊥ . The perpendicular jump frequency, 𝜅⊥ is evaluated (from Figures 9 and 10) by inversing the time required for 50% of the molecules to get displaced by a distance not less than 𝜆⊥ .

An increase in εls results in an increase of the inter-atomic attractive forces between the water molecules and the substrate, thereby reducing the rate of displacement of the molecules close to the solid-liquid interface (Figure 9). However, it can be seen from Figure 10 that the rate increases with the rise in temperature. Further, it can be observed (Figures 9 and 10) that the molecular displacement frequency in the parallel direction is 24 ACS Paragon Plus Environment

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about twice that of the perpendicular direction (𝜅∥ ∼ 2𝜅⊥ ) and is attributed to the additional degree of freedom in the x-y plane.49

To understand the bulk movement of water molecules during the droplet spreading, the bulk frequency (𝜅𝐵𝐵𝐵𝐵 ) has also been evaluated using similar approach as adopted by Ruijter et

al.53 (See Supplementary Information 6). However, the contact line movement is governed by the water molecules located at the first two layers. The frequency of parallel jumps is calculated to be much higher than that of the perpendicular jumps. Thus, majority of the molecules undergo lateral displacements at any given time, and the evaluated values of 𝜅∥

are used for computing the three-phase contact line friction coefficient, ζ53,49 at different values of εls and temperature (Figure 11).

Figure 11 Variation of ζ as a function of temperature and ε (From MD simulations of droplet spreading). Increases in the values of εls denote enhanced attractive forces between the droplet and the substrate leading to increased hydrophilicity and contact line friction. As a consequence, 25 ACS Paragon Plus Environment

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significant increase in the contact line friction coefficient, ζ, with increase in εls is observed. The results also show appreciable reduction in ζ with temperature. It is clear from equation 3 that the contact line friction is a function of temperature and the frequency of the jumps of the water molecules, κ0. With increase in temperature, the kinetic energy of molecules increases, raising the frequency of jumps of the water molecules. It has been observed that a 4 % increase in temperature from 298 K to 310 K leads to a 60.9 % increase in the frequency of jumps, at εls=6.54. Thus, the amplification in the frequency of the molecular displacement with temperature more than compensates for the increase in temperature and results in the decrease of ζ (a net decrease of 34.74 % over the entire range of temperature for εls=6.54). 4.4 Comparison with the theoretical model The contact line friction coefficient, ζ is a strong function of both the surface wettability and substrate temperature. The values of ζ for each constant wettability substrate and at specific temperatures are evaluated from the MD simulation of the spreading of sessile droplets as described in sub-section 4.3 and presented in Figure 11. Using the ζ values corresponding to the known values of εls (Figure 11) at different x-locations, a spatial average of ζ is evaluated from the simulations of droplet spreading on substrate 3. These spatially averaged ζ values (at different temperatures) are compared with the ζ values obtained from the theoretical model and are presented in Figure 12. It is clear from the figure that the values of ζ and their variations, as obtained from these two approaches, are quite close. A recent experimental study has reported qualitatively similar nature of variation.16 However, the results from that study cannot be directly compared with those of the present MD study since the scale of the system simulated herein is several orders of magnitude smaller than that encountered in the experiments (millimeter).24 26 ACS Paragon Plus Environment

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Figure 12 Comparison of ζ between the simulation and theoretical studies The results from the MD simulations provide valuable insights in envisaging the droplet movement with thermal fluctuations and the interfacial energy gradient. The information will be useful in the design of functional surfaces where the directional droplets movement can be augmented at elevated temperatures.

5. Conclusion Molecular dynamic simulations of wettability gradient induced water droplet motion on a solid substrate are performed using GROMACS and Visual Molecular Dynamics (VMD). The motion of the droplet is a result of the unbalanced surface tension forces at the nonequilibrated contact line. The gradient surface is created using different values of the interaction parameter to mimic reported experimental trends in droplet motion [with a maximum velocity of the order of 3 m/s (for substrate 1) to 4 m/s ( for substrate 3)]. The simulation results demonstrate a distinct increasing trend of the droplet movement with 27 ACS Paragon Plus Environment

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temperature, for example, an average increase of ~26% in the droplet velocity is observed for a 10 K rise in the temperature). This increase is attributed to the decrease in the contact line friction wherein the relevant coefficient, ζ decreases from 0.0052 Pa.s to 0.0035 Pa.s. The proposed model successfully explains the droplet movement in terms of the wettability gradient induced driving forces and the relevant resistive forces acting on the droplet. The predictions from the MD simulations and the theory are successfully compared including the variations of ζ with temperature.

Acknowledgements The authors gratefully acknowledge the financial support provided by the Indian Institute of Technology Kharagpur, India [Sanction Letter no.: IIT/SRIC/ATDC/CEM/2013-14/118, dated 19.12.2013].

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Greaphical Abstract

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Molecular Dynamics Study of Thermally Augmented Nanodroplet

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