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Free Energy and Dynamics of Water Droplet Coalescence Chi Yuen Pak, Wentao Li, and Ying-Lung Steve Tse J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b06507 • Publication Date (Web): 17 Sep 2018 Downloaded from http://pubs.acs.org on September 21, 2018

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The Journal of Physical Chemistry

Free Energy and Dynamics of Water Droplet Coalescence Chi Yuen Pak, Wentao Li, and Ying-Lung Steve Tse* Department of Chemistry, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong SAR, China.

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ABSTRACT

Droplet coalescence is a critical issue in the atmospheric sciences. In this work, the coalescence of water nanodroplets was studied by performing equilibrium and nonequilibrium molecular dynamics simulations. To understand the intrinsic nature of the process, we obtained the free energy change as a function of droplet size and droplet-droplet distance. We decomposed the free energy change ΔF into energetic ΔU and entropic TΔS contributions to understand the molecular details. ΔU was dominated by the change in Coulomb interactions, which strongly correlated with the change in the number of hydrogen bonds. We found a strong positive correlation between the mobility of water molecules and TΔS. To analyze the dynamics, two colliding water droplets of the same size were given different initial speeds, impact parameters, and collision angles. We found when the collision is head-on, the time for thorough mixing between interfacial and bulk molecules decreases when the initial speed increases, whereas when the collision is off-center, the induced torque significantly increases the mixing time, which can last up to hundreds of picoseconds. The initial impact of a collision can push the interfacial molecules away from the center of mass, and provide an evaporation mechanism of the interfacial/adsorbed molecules on the droplet.


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1. INTRODUCTION The consequences of collisions between droplets have received great attention in recent years due to their importance in both atmospheric1–6 and industrial applications.7–13 Investigations, by either experiments3,7,14,15 or numerical calculations,16–21 were conducted in the last few decades by different research groups. Different outcomes of droplet collisions were studied in terms of “stretching separation”, “shattering collision”, and “coalescence” in the literature.3,7,14,15,17–23 In the stretching separation studies, two droplets first combine but are later separated, whereas in shattering collision studies, two droplets collide and then “explode”. The conditions for the different consequences, in terms of factors such as velocity, impact parameter, were systematically investigated. Coalescence, as the main topic in the present study, is a process in which two or more particles/droplets merge to form a larger one permanently. There have been experimental and simulation work that examined coalescenceinduced jumping that can occur when two nanodroplets come together on a superhydrophobic surface.24–28 It was concluded that the jumping motion was caused by the impact force acting on the superhydrophobic surface when the bridge that connects the coalescing nanodroplets expands during coalescence. In addition to the outcomes of collision dynamics of water droplets, the previous studies also focused on the coalescence of water droplets as it is related to the growth of cloud droplets and raindrop formation.1,3–6 It is believed that the development of cloud droplets is not only driven by condensation of water vapor, but also by droplet coalescence in the atmosphere.4,6,29–32 Therefore, the distribution of droplet sizes in clouds, which is primarily controlled by the coalescence between droplets and their breakup, is also a critical issue to study.29,31–35 Previous experimental work systematically examined the binary collisions of water droplets and studied how the fate of a collision, whether it is coalescence or shattering/separation, depends on the Weber numbers of the droplet pair and their size ratio.14,22,23 Experiments on binary


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coalescence of liquid micro-droplets were also carried out with the aid of advanced imaging tools36–39 as well as optical tweezers.40 As computing power has improved tremendously in recent decades, computers are now important tools for understanding droplet collisions that lead to coalescence or shattering of droplets. Early numerical modeling and calculations were carried out and reported by Greenspan and Heath to study the modes of colliding microdroplets of water.16 Later, Svanberg et al. performed numerical calculations on the consequences of collisions of equal-sized nanoscale droplets under different velocities and impact parameters systematically and found good agreement with experiment.18 In 2006, Rekvig and Frenkel reported a mechanism for oil droplet coalescence in water with and without surfactant by dissipative particle dynamics (DPD) simulations.41 Not only numerical and DPD simulations have been used to study droplet coalescence, molecular dynamics (MD) simulations have also played an important role in elucidating the molecular mechanisms of coalescence at molecular scales. In most MD studies, dynamical runs beginning from nonequilibrium configurations were performed to study the dynamics of coalescence of either water42–45 or metal nanodroplets.46–49 Zhao and Choi reported the coalescence process of equal–sized nano-scale water droplets, in which there were 100 water molecules per droplet in both vacuum and n-heptane cases.42 In 2007, Liao, Ju, and Yang studied the temperature and size effects on the coalescence of water nanodroplets.43 More recently, electrocoalescence of water nanodroplets with dissolved KCl has been studied by Wang et al.44,45 In addition to the studies of the coalescence of water droplets, the coalescence of silver nanoparticles,46 TiO2 nanoparticles,47 and ZnSe nanoparticles48 were also reported. In the studies of free energy profiles of bimetallic nanoparticles, Farigliano et al. used twodimensional well-tempered metadynamics to reconstruct a free energy profile of the coalescence of cobalt (Co) and gold (Au) nanoparticles as a function of the distance between


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the two centers of mass of clusters and the radius of gyration.50 This free energy profile was compared with a “pseudo” free energy profile that they computed from an independent set of 242 dynamical runs of the (irreversible) coalescence process.49 These previous studies demonstrated that the free energy change of coalescence can provide important insights for the process. In this paper, we systematically study the free energy profiles, also commonly called potentials of mean force, of the coalescence of pure water droplets as a function of system size. To understand the molecular details of coalescence, the free energy changes were further decomposed into energetic and entropic contributions. The potential energy changes of the process were investigated in terms of the number of hydrogen bonds, and the entropic change was compared to the mobility of water. Finally, the dynamics of the collision between water droplets was studied by nonequilibrium simulations. We investigate the fates of droplet collision in terms of different initial conditions including speed, collision angle, and impact parameter. For the collisions that resulted in coalescence, we studied the change in the number of hydrogen bonds as well as the mixing progress of interfacial and bulk water molecules after collision. In section 2, the simulation details are presented. In section 3, we discuss the results and analyses, including the free energy profiles, analyses on the energetic and entropic components, results and analyses of collisions in the nonequilibrium systems. Finally, we give conclusions in section 4. 2. SIMULATION DETAILS 2.1 Equilibrium simulations. All-atom molecular dynamics (MD) simulations were carried out using GROMACS v5.1.4.51 The water model used was the SPC/E model,52 and the SETTLE algorithm53 was used to constrain the water bond lengths and angles. The combination rules for the Lennard-Jones (LJ)


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interactions were the Lorentz-Berthelot combination rules. The equations of motion were integrated using the leap-frog algorithm. The simulations were carried out in the canonical (NVT) ensemble (constant number of particles, volume, and temperature) at 300 K with a coupling time constant of 0.1 ps through the velocity-rescaling algorithm by Bussi, Donadio, and Parrinello.54 There were five different setups containing droplets of five different sizes. Each of the simulations consisted of a pair of water droplets. One droplet, called the “left/L” droplet, was always made of 1000 water molecules, whereas the other droplet, called the “right/R” droplet, was made of 100, 250, 500, 750 or 1000 water molecules. We have labeled these droplets as “100w”, “250w”, “500w”, “750w”, and “1000w” below. The simulation box was nonperiodic without any cut-offs for the interactions, which were reported to cause problems in interfacial systems.55 To prevent atoms from leaving the simulation box, a flatbottomed restraint potential was applied in each dimension, whose details are provided in the Supporting Information (SI). The reaction coordinate along which the free energy profiles were calculated is defined as follows. In each simulation, we first divided the simulation box into the left side and the right side by drawing a dividing line midway between the two droplet centers. The x coordinate of this dividing line is C. Since each water is either on the left side (L) or the right side (R) of this division line, in order to ensure the centers of mass to change continuously, we introduce a switching function SA(x), where A is L or R, as follows:

if x  C 1  SL ( x)  1  (C  x) 2 [2(C  x)  3w] / w3 if C  x  C  w 0 if x  C  w 

(1)

SR (C  x)  SL (C  x)

(2)

where w was set to be 0.05 nm. The definition of SR is a reflection of SL about the line x = C. More details about the switching functions are shown in the SI.


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The centers of mass COMA (A = L or R) are then defined as: N

COM A   SA ( xi )mi xi i 1

N

S i 1

A

( xi )mi

(3)

where N is the total number of atoms, mi and xi are the mass and the coordinates of the ith atom. The free energy profiles were calculated by the umbrella sampling method56 using our modified version of PLUMED57 v2 and the weighted histogram analysis method (WHAM).58,59 The reaction coordinate of the free energy profile was the distance in the x direction between the COMs of the two droplets. The reaction coordinate was restrained from 13.5 to 75 Å in a total of 124 windows evenly spaced windows. The restraining potentials used for the different windows and different sizes of the R droplet are provided in Table S1 in the SI. For equilibration, the droplets were first relaxed at 300 K with a time step of 1 fs for at least 2 ns with a separation distance of 75 Å between the two droplets. To obtain the initial configurations of the other windows, the R droplet was pulled towards the COM of the L droplet along the reaction coordinate x. The distance between two COMs was restrained at the desired value in each window, and the system was further relaxed at 300 K with a time step of 2 fs for 2 ns before collecting production data for another 2 ns. We sampled four special windows at different values of the reaction coordinate (x distance between the two COMs) for 20 ns to improve statistics for some of the analysis presented in section 3. The separation distances in these four special windows were (1) 70 Å, when two droplets are well separated from each other, (2) when two droplets are just starting to coalescence, (3) 25 Å, when two droplets are about half combined, and (4) when two droplets have combined to form one large droplet. The exact separation distances for (2) and (4) depended on the size of the R droplet; this information is provided in Table S2 in the SI. Snapshots of these four special windows for R droplet of 1000 water are shown in Figure 1.


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Figure 1. Snapshots for the four special windows in the 1000w case. These represent four important stages during coalescence: (a) two separate droplets, (b) coalescence has just started, (c) coalescence is half-way finished, and (d) coalescence is finished.

2.2 Nonequilibrium simulations. To understand the dynamics of coalescence, two water droplets each containing 1000 water molecules were brought together with various initial conditions. The initial conditions considered are speed v, impact parameter b, and collision angle  as shown in Figure 2.

Figure 2. Schematic diagram showing the initial conditions of a collision simulation of two droplets with speed v, (i) impact parameter b (defined to be the y-distance between the COMs of the droplets), and (ii) collision angle θ. At first, the two COMs were separated by 60 Å in the x direction and by b in the y direction by harmonic restraints with force constant 2.39 kcal/(molÅ2). One run was first equilibrated in the canonical (NVT) ensemble at 300 K for 5 ns with a time step of 2 fs. From the last 4 ns of the NVT run, we sampled 100 configurations that were to be used as the initial states of the


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nonequilibrium simulations. To ensure the dynamics would be consistent with classical mechanics, these 100 configurations were then propagated with constant NVE dynamics with a time step of 2 fs. To better conserve the total energy, double precision was used in these simulations. At the start of each NVE run, we removed the restraint to allow the collision to happen. Moreover, velocities with magnitude v and directions determined by the collision angle

 were added to the original velocities of all the atoms. The time for each production run was 500 ps with time step of 2 fs. In all cases, the droplets came in full contact within 10 ps, but we allowed these simulations to run for at least 500 ps to capture the aftermath of the collision. To avoid any strong effects caused by the walls of the simulation box, at the beginning of each run, the xyz-dimensions of the box were extended and controlled by the flat-bottomed restraint potential as given in the SI. The initial conditions are given in Table 1 in section 3.2. 3. RESULTS AND DISCUSSION 3.1 Equilibrium (quasi-static) simulations. 3.1.1 Free energy profiles. A free energy profile ΔF as a function of the distance between the two droplets allows us to study the intrinsic driving force of the process. We first calculated ΔF as a function of the x distance between the two COMs (see section 2.1) as shown in Figure 3a. In these free energy profiles, we fixed the number of water molecules in the L droplet at 1000, while we varied number of water molecules in the R droplet from 100 to 1000. The x coordinate of the free energy minimum is labeled xmin, and we set ΔF(xmin) = 0. We show a way to predict the xmin value in the SI.


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Figure 3. (a) Free energy profile as a function of the x distance between the two COMs and the size of the R droplet (size of L droplet is fixed). Error bars were calculated by bootstrapping with an assumed correlation time of 10 ps. The x coordinate of the free energy minimum is denoted by xmin. ΔF(xmin) was set to be 0. (b) ΔFcoal = –ΔF(∞) is shown as a function of the reciprocal of the total number of water molecules 1/Nw. The linear equation of the fit is shown in the plot. The y-intercept represents the free energy change for the L droplet (of 1000 water molecules) to coalesce with an infinitely large R droplet.

When two droplets are separated far from each other, the free energy curve remains flat until the distance between the two COMs is down to about 55 Å, if the R droplet is made of 1000 water molecules. When the R droplet contains fewer molecules, the free energy curve remains flat until the COM distance is smaller (the R droplet containing 100 water molecules, for example, has a free energy that is flat until 40 Å). When the two droplets come close enough, coalescence starts and ∆F decreases, indicating the process is spontaneous. For x < xmin, as the spherical shape of the combined droplet is deformed, ∆F(x) > ∆F(xmin) = 0. We adopt the notation ΔFcoal to be the change in ΔF (or ΔΔF) from the state when the two droplets are totally separated (x→∞) to the state at the location of the free energy minimum xmin. Explicitly, ΔFcoal ≡ ΔF(xmin) – ΔF(∞) = 0 – ΔF(∞). For the different free energy curves in Figure 3a, the general features are similar, but when the size of the R droplet increases, ΔFcoal becomes more negative. In fact, Figure 3b shows there is a clear linear relationship between ΔFcoal and the reciprocal of the total number of water molecules 1/Nw. The y-intercept of the


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linear fit, at which 1/Nw equals 0, provides an estimate for the free energy change when the L droplet (of 1000 water molecules) coalesces with an infinitely large R droplet (Nw→∞). These free energy profiles will be further analyzed by the decomposition of ΔF into its energetic and entropic contributions next. As a side note, the free energy change of coalescence ΔFcoal describes a reversible process and nonequilibrium statistical mechanics provides a lower bound for the average work done to the surroundings:60,61 ΔFcoal ≤ ⟨W⟩, where ⟨W⟩ is the work done to the surroundings averaged over different initial microstates of the system. (Note: For our choice of sign convention, energy change from the system to the surroundings is negative). Furthermore, the framework described here can be easily extended to study the coalescence involving more complicated droplets that include ions and/or coatings whose effects are actively studied in the atmospheric sciences.62–64 3.1.2 Potential energy and hydrogen bonds. One can gain insight into how molecular details affect the free energy change of coalescence by studying the potential energy change ΔU. In order to improve the statistics of estimating ΔU, we first divide the two droplets into four regions as 1) left bulk “LB”, 2) left interface “LI”, 3) right bulk “RB”, and 4) right interface “RI” as shown in Figure 4a.65–68


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Figure 4. (a) Schematic diagram showing the regions of left bulk (LB), left interface (LI), right bulk (RB), and right interface (RI). (b) Contour plot showing the Gaussian mass wave density of water molecules. In this particular configuration, the two droplets have started merging. For the contour plots of the other special windows and R droplet sizes, see the SI.

In our calculations, the instantaneous interfaces for each frame are defined using the algorithm by Willard and Chandler,67,69,70 as shown in Figure 4b. Similar to ΔF, ΔU(xmin) was set to be 0. The intermolecular pair interaction energies between two water molecules in terms of Coulomb (UCoul) and LJ (ULJ) are calculated in each of the four regions (LB, LI, RB, and RI) as a function of x. Furthermore, the number of water molecules n as a function of x is tracked in each region. The total intermolecular interaction energy for the system Usys is simply the sum of the number of such pairs multiplied by their corresponding pairwise interaction energies.68 As a result, the change in the system potential energy ΔUsys can be calculated by the following relations: U sys ( x)U Coul ( x)U LJ ( x)

U Coul ( x)



n( x )

a{LB,LI,RB,RI}

U LJ ( x)



a{LB,LI,RB,RI}

n( x )

a

a

(4)

ECoul ( x) ELJ ( x)

a

a

(5) (6)

where the subscript a represents one of the four regions (“LB”, “LI”, “RB”, or “RI”). The angular brackets ⟨·⟩a represent averaging over all pair interactions of the same type in region a. Usys is the sum of the overall intermolecular energies in the four regions. Direct calculations of the intermolecular energies usually have large statistical uncertainty, but the averaging carried out in Eqs. (5) and (6) significantly reduces the statistical uncertainty. Furthermore, as mentioned in section 2.1, we collected significantly more data at the four special droplet separation distances to further reduce the uncertainty. The changes of these energies as a function of x distance separation of the COMs are shown in Figure 5. Similar trends for the energy changes can be observed for the different R droplet sizes. The trend in ΔUCoul is largely


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responsible for the overall trend in ΔUsys, whereas ΔULJ hardly contributes to ΔUsys. In other words, the main contribution to the change in the system energy is the Coulomb interactions. In particular, ∆UCoul generally decreases from the state of total separation of the droplets to the final combined state, and this indicates that the driving force of the coalescence is dominated by charge-charge interactions. As coalescence begins (from larger x to smaller x in Figure 5), ΔUCoul and ΔUsys rise slightly before they eventually decrease as the separation of droplets decreases. This feature can be understood by studying the variation in the total number of hydrogen bonds nHB in Figure 5 as discussed below.

Figure 5. Changes in Coulomb interaction energy, ∆UCoul, Lennard-Jones interaction energy, ∆ULJ, system potential energy, ∆Usys, and the negative (for cleaner comparisons with ∆UCoul and ∆Usys) of the total number of h-bonds −nHB as a function of the x distance between the droplets. To put the −nHB and ∆UCoul curves in the same plot, the scales were chosen by minimizing the sum of the squared differences between the points. Only the 1000w case is shown. For the other R droplet sizes, see the SI. In our hydrogen bond (h-bond) analysis, two water molecules are defined as hydrogen bonded when the distance between the two oxygen atoms is less than 3.5 Å and the O—H···O angle is less than 30°.71–73 In Figure 5, we show the negative of the total number of h-bonds −nHB for the whole system as a function of the x distance between droplets. We can clearly see that ΔUCoul and −nHB closely resemble each other. The clear resemblance implies the change in the Coulomb energy (or the total system energy) is a direct consequence of the change in the total number of h-bonds. The slight increase in ΔUCoul from x = 70 to 54 Å was due to the increase in the surface area as the peanut shape was formed as shown in Figure 4a. The


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interfacial water molecules do not form as many h-bonds as the bulk water molecules so that when nHB decreased (–nHB increased), then ΔUCoul increased. As x decreased further from 54 Å, the two droplets continued to merge to form a sphere, for which the number of interfacial water molecules is minimized. The decrease in the number of interfacial water molecules led to an increase in nHB (decrease in –nHB) and a decrease in ΔUCoul (See the SI). To see how the change in nHB depends on the droplet size, as in ∆Fcoal (Figure 3b), we first define ΔnHB ≡ nHB(xmin) – nHB(∞). When the R droplet size increases (1/Nw decreases), ΔnHB increases. In fact, there also exists a linear relationship between ΔnHB and the reciprocal of the total number of water molecules 1/Nw as shown in Figure 6a. Because ∆Fcoal and ΔnHB are both linearly linear in 1/Nw, they are linearly related to each other as shown in Figure 6b. From the slope, we estimate that the average free energy of one h-bond contributes about −2.7 kcal/mol to ∆Fcoal. This value is rather close to the Gibbs free energy change of h-bond formation at −2.9 kcal/mol using the estimated ∆H and ∆S, which were based on experimental data, at 300 K by Suresh and Naik.74

Figure 6. (a) Change in the total number of h-bonds ∆nHB as a function of the reciprocal of the total number of water molecules 1/Nw. (b) Free energy change ΔFcoal as a function of ∆nHB.


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3.1.3 Entropic contribution and mean squared displacement. Direct calculation of an entropic contribution T∆S is difficult, but it can be readily deduced from the relation T∆S = ∆U − ∆F when ∆F and ∆U are both available. Using the information from Figure 3a and Figure 5 (See the SI for the other R droplet sizes), the entropic contributions are shown in Figure 7a. T∆S generally decreases from the state when the two droplets are separated to the state when coalescence is finished. This is consistent with the fact that the number of assessable configurations, which ∆S measures, becomes fewer after the droplets have merged. Alternatively, ∆S can be loosely related to the “randomness” of the molecules, and this can be linked to the water dynamics during coalescence.

Figure 7. (a) Free energy changes ΔF and decompositions of ΔF into energetic ΔUsys and entropic contributions TΔS, (b) MSDs of water and entropic contributions for different R droplet sizes (250w, 500w, and 1000w) as a function of the x distance between droplets. To put the MSD and TΔS curves in the same plot, the scales were chosen by minimizing the sum of the squared differences between the points. For the 100w and 750w cases, see the SI. The mean squared displacement (MSD) of time interval Δt of water can be calculated by

1 1 N w Nt 2 MSD(t )   ri (tk  t )  ri (tk ) N w Nt i 1 k 1

(7)

where ri(t) is the position of the water molecule i at time t, Nw is the total number of water molecules, and Nt is the number of times of shifting the time origin used to improve the statistics (this is possible as the simulation in each window is quasi-static with the biased umbrella


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potential). Δt was chosen to be 0.4 ps in this analysis. If the MSD of water is larger, the water molecules have a higher degree of fluctuations (more randomness) in the positions, so T∆S should increase. Therefore, we expect the MSDs and the entropy contributions to be positively correlated. The MSDs of water for different R droplet sizes as a function of the x distance between the two COMs were calculated and plotted together with T∆S in Figure 7b, which clearly shows the MSD and T∆S are positively correlated (we observed the same correlation for larger intervals such as 0.8 ps, but a larger time interval increases the uncertainty in the MSD). The MSDs of water generally decrease from the total separation of the droplets to the end of coalescence. This decrease in the MSD is strongly correlated with an increase in the total number of h-bonds (see the SI). As the water molecules are more strongly connected to each other, the fluctuations in the water positions and the MSDs are reduced. 3.2 Coalescence as an irreversible process. Studying the coalescence of water droplets quasi-statically (reversibly) allowed us to directly calculate the free energy profiles. In practice, the coalescence/collision of water droplets is an irreversible (nonequilibrium) process. To gain insight into the time dependence and the molecular features of the nonequilibrium process, we carried out a series of constant NVE collision simulations, which preserve classical mechanics, with different initial conditions for the initial speed v, the impact parameter b, and the collision angle θ for the two droplets as shown in Table 1 below. Table 1. Initial conditions for the nonequilibrium simulations. case a b c d e

v (m/s) b 270 530 790 1000 1300 0 270 530 700 980 1400 r† 270 500 1100 1700 2400 1.5r 270 530 850 1200 1500 0 270 530 1200 1900 2600 0 † r = radius of the 1000w droplet = 19 Å


 0° 0° 0° 30° 60°

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3.2.1 Fate of collision. When two water droplets come together at different angles, impact parameters, and initial speeds, different fates can happen. Two water molecules are said to be in the same fragment if the O-O distance is less than 3.5 Å (the radius of the first solvation shell). Based on previous work by Svanberg et al.18,75 and Kalweit et al.,20 we label the outcome of each run using the sizes of the largest two fragments at simulation time t = 500 ps as below: i) If the largest fragment has 70% of all the water molecules, the outcome is identified as “coalescence”. Otherwise, the outcome is identified as “separation” that is further classified below. ii) For separation, if the sum of the two largest fragments have more than 80% of all water molecules, the outcome is identified as “stretching separation”. If less than 80%, the outcome is identified as “shattering”. Based on the classifications above, we have summarized our results for all of the five cases below in Figure 8.

Figure 8. (a) Outcomes for cases a to c in Table 1 as a function of initial speed v and impact parameter b. The inset shows the induced rotation of a water fragment with torque. (b) Outcomes for cases d and e as a function of v and collision angle . At each set of the initial conditions, all 100 runs all gave the same outcome.

When the collision was head-on (b = 0), coalescence for the two droplets happened until the initial speed v reached 1300 m/s. When the collision was off-center (b > 0), a torque was


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induced as shown in the Figure 8a inset, but no separation was observed until v reached 700 m/s for b = r (500 m/s for b = 1.5r); at this speed, the two rotating fragments eventually flew away from each other and from their rotation axis. At v = 1400 m/s for b = r (2400 m/s for b = 1.5r), shattering started to happen as small fragments formed. Note that coalescence changes to stretching separation at a smaller v as b increases. This is consistent with the previous study.2,14,18 We also found that stretching separation changes to shattering separation at a larger v as b increases. When the two droplets came together at collision angle , their speeds in the x and y directions were vx = ±v cos and vy = v sin. It may appear that the situation in case d is the same as case a in which  = 0° (and b = 0) because shattering in both cases started happening at vx = 1300 m/s, but the differences will become clearer in a later subsection. Specifically, the mixing in the z direction is dependent on both vx and vy, the latter of which was not zero in cases d and e with  > 0°. 3.2.2 Change in the total number of hydrogen bonds. For brevity, only case a is discussed here. The results are similar for the other cases. Initially, for the two well-separated 1000w droplets, the total number of h-bonds nHB was around 3350 as shown in Figure 9a. As the initial speed was added to the droplets, nHB dropped as the shape of the spherical droplets deformed. The initial decrease in nHB was more significant as v was increased. At the higher speeds (≥ 790 m/s), the droplets became ellipsoids or even disks at the highest speed. Therefore, our results can be generally classified into two sets: one with initial v smaller than 790 m/s and another with v larger than 790 m/s. For the results with v smaller than 790 m/s, nHB basically remained constant after coalescence. On the other hand, for the droplets with initial v larger than 790 m/s, on the other hand, nHB temporarily rose as coalescence began and a more spherical object with a larger bulk volume formed. Subsequently after the increase in nHB, these curves then started their descent as shown in the sloping tails in Figure 9a. This gradual


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decrease in nHB was caused by the boiling/evaporation of water molecules as the instantaneous temperature remained high as shown in Figure 9b. Even well after initial collision, the final values of nHB for each case were different due to the different temperatures.

Figure 9. (a) Total number of h-bonds nHB and (b) instantaneous temperature as a function of time shown for each initial droplet speed, v. The instantaneous temperatures at t = 0 for v = 790, 1000, 1300 m/s are not shown; they were 530, 640, and 910 K, respectively.

3.2.3 Mixing in coalescence. We studied how the mixing of water molecules proceeded during coalescence by again dividing the system into the interfacial (LI + RI) and bulk (LB + RB) regions as previously shown in Figure 4. To track how the water molecules initially in these two regions moved as coalescence happened, we define “mixing” functions m(τ) by calculating the average of a squared coordinate (x, y, or z) of the water molecules in the largest fragment in region a as a function of time τ: mx,a(τ) = ⟨x(τ)2⟩a /⟨x(τ)2⟩all

(8)

where a is interfacial (I) or bulk (B), and “all” stands for I and B. Definitions for my,a(τ) and mz,a(τ) are analogous to mx,a(τ). These m(τ) functions have the feature that when their value stabilizes at 1, it indicates that thorough mixing has happened in the studied directions as memory of the history disappears. Since the time it took for each run to start the coalescence process would be different, we need to find a common reference time to study the mixing progress for all runs. τ = 0 for each run is


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defined as the time at which the COM distance first becomes ≤ 2r. The averaging in m(τ) was carried out across all runs with this τ = 0 definition for each run. We note in this analysis that once the water molecules are identified initially in either I or B region, they carry the same I or B label throughout the entire analysis for the sake of tracking the progress of mixing in each run. We show some of the results for cases a, b, and d in Figure 10.

Figure 10. Mixing progress expressed in terms of the m(τ) functions. If a m(τ) function stabilizes at 1, it is an indication that mixing is complete in that dimension(s). The initial conditions are shown at the top of each plot. In plots i to iii, since the y and z directions are identical, the results were obtained by averaging the two directions. Plots i to iv share the same legends, but plot iv has the additional legend for mz.

For case a (Figure 10i and ii), the I curves approached 1 from above because these water molecules were initially at the interface, while the B curves approached 1 from below as the


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water molecules were from bulk. Furthermore, as v increased (Figure 10ii), the time for the m curves to reach 1 decreased, implying faster mixing. mx,I(τ) had a sharp increase that was quickly followed by a sharp decrease right after a head-on collision. Similarly, a sharp decrease then increase for the mx,B(τ) happened right after the collision. These features were caused by the initial compression of the two droplets in the x direction when the two droplets collided. Soon after, the combined droplet relaxed back to a spherical shape. The moment when the combined droplet first became a sphere did not mean mixing was complete as the m(τ) curves had not all yet returned to 1. For case b, in which the collision was off-center (Figure 10iii), the mx and my curves are oscillatory and out of phase due to rotational motion. The mx curves also carried the same features as case a where the mx curves first increased and decreased due to compression. All these m curves never truly stabilized to 1 as memory of the collision lingered for significantly longer than 500 ps. In all the studied cases, mixing took at least 70 ps to finish after the initial collision. For case d (Figure 10iv) in which the two droplets came together at collision angle  = 30°, the collision speed in the x direction vx = v cos = 1000 m/s was the same as that of Figure 10ii. One might think the dynamics in the z direction in Figure 10iv would be identical to that of y in Figure 10ii since z in the former case and y in the latter case were perpendicular to the collision directions. However, the mz(τ) functions in Figure 10iv are clearly different from the my(τ) functions in Figure 10ii. This demonstrates that mixing in the x, y, and z directions can be coupled through molecular interactions; a collision in the x direction can affect mixing in the other two directions, vice versa. In all the plots in Figure 10, right after the collision, mx,I increased sharply. This implies that the collision moved interfacial molecules on average further away from the droplet center after the initial impact. This outward motion can promote evaporation of these interfacial molecules,


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and have implications for the escape of the interfacial water molecules and/or any adsorbed molecules that are common in the atmosphere.68 4. CONCLUSIONS Coalescence is crucial for understanding the growth of atmospheric water droplets and the formation of raindrops. We discovered a linear relationship between Fcoal and the net change in the total number of h-bonds before and after coalescence nHB. The free energy change of h-bond formation derived from this linear relationship agrees reasonably well with the result of Suresh and Naik.74 We further decomposed each F into its energetic U and entropic TS components. We showed the change in nHB is the primary cause for U. We observed a strong positive correlation between the trends in entropic contributions TS and the MSDs of water molecules. The rationale is that the larger the MSD, the more “random” the positions of water molecules will be, and so will TS. We carried out a series of nonequilibrium runs to directly probe the fates of two colliding droplets at different initial conditions (speed, impact parameter, and collision angle) and analyzed all the observed fates in terms of coalescence, stretching separation, or shattering. In particular, we analyzed the mixing progress in each coalescence situation. Even though higher initial collision speeds resulted in faster mixing, it still took at least 70 ps in all our cases after the initial collision before mixing can be considered finished. In some cases, the memory of an off-center collision lingered much longer when a rotation was induced, causing the mixing time to last up to hundreds of picoseconds. Furthermore, the initial impact of the collision pushes the interfacial molecules further outwards from the droplet, and this has implications for the evaporation of adsorbed gaseous impurities. The free energies of sorption for many common gaseous pollutants by water are less than 10 kcal/mol,68,76,77 which can be readily provided by the energy transfer due to a collision between droplets.


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AUTHOR INFORMATION Corresponding Author *E-mail: [email protected] ACKNOWLEDGMENT This research was supported by the Direct Grants (4053156, 4053198, and 4053275), the Faculty Strategic Fund for Research from the Faculty of Science at The Chinese University of Hong Kong, and the Research Grants Council of Hong Kong (CUHK 24301817). S.T. acknowledges the research start-up allowance by the Croucher Foundation. ASSOCIATED CONTENT Supporting Information Further details of simulations and additional plots for other droplet sizes. REFERENCES (1)

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Figure 1. Snapshots for the four special windows in the 1000w case. These represent four important stages during coalescence: (a) two separate droplets, (b) coalescence has just started, (c) coalescence is half-way finished, and (d) coalescence is finished. 54x35mm (300 x 300 DPI)



Figure 2. Schematic diagram showing the initial conditions of a collision simulation of two droplets with speed v, (i) impact parameter b (defined to be the y-distance between the COMs of the droplets), and (ii) collision angle θ. 57x54mm (300 x 300 DPI)


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Figure 3. (a) Free energy profile as a function of the x distance between the two COMs and the size of the R droplet (size of L droplet is fixed). Error bars were calculated by bootstrapping with an assumed correlation time of 10 ps. The x coordinate of the free energy minimum is denoted by xmin. ∆F(xmin) was set to be 0. 64x52mm (300 x 300 DPI)



Figure 3. (b) ∆Fcoal = –∆F(∞) is shown as a function of the reciprocal of the total number of water molecules 1/Nw. The linear equation of the fit is shown in the plot. The y-intercept represents the free energy change for the L droplet (of 1000 water molecules) to coalesce with an infinitely large R droplet. 65x53mm (600 x 600 DPI)


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Figure 4. (a) Schematic diagram showing the regions of left bulk (LB), left interface (LI), right bulk (RB), and right interface (RI). (b) Contour plot showing the Gaussian mass wave density of water molecules. In this particular configuration, the two droplets have started merging. For the contour plots of the other special windows and R droplet sizes, see the SI. 84x104mm (300 x 300 DPI)



Figure 5. Changes in Coulomb interaction energy, ∆UCoul, Lennard-Jones interaction energy, ∆ULJ, system potential energy, ∆Usys, and the negative (for cleaner comparisons with ∆UCoul and ∆Usyss) of the total number of h-bonds −nHB as a function of the x distance between the droplets. To put the −nHB and ∆UCoul curves in the same plot, the scales were chosen by minimizing the sum of the squared differences between the points. Only the 1000w case is shown. For the other R droplet sizes, see the SI. 52x32mm (300 x 300 DPI)


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Figure 6. (a) Change in the total number of h-bonds ∆nHB as a function of the reciprocal of the total number of water molecules 1/Nw. 70x60mm (300 x 300 DPI)



Figure 6. (b) Free energy change ∆Fcoal as a function of ∆nHB. 70x60mm (300 x 300 DPI)


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Figure 7. (a) Free energy changes ∆F and decompositions of ∆F into energetic ∆Usys and entropic contributions T∆S. For the 100w and 750w cases, see the SI. 65x53mm (300 x 300 DPI)



FIgure 7. (b) MSDs of water and entropic contributions for different R droplet sizes (250w, 500w, and 1000w) as a function of the x distance between droplets. To put the MSD and T∆S curves in the same plot, the scales were chosen by minimizing the sum of the squared differences between the points. For the 100w and 750w cases, see the SI. 65x53mm (300 x 300 DPI)


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Figure 8. (a) Outcomes for cases a to c in Table 1 as a function of initial speed v and impact parameter b.At each set of the initial conditions, all 100 runs all gave the same outcome. 65x53mm (300 x 300 DPI)



Figure 8. (a) The inset shows the induced rotation of a water fragment with torque.


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Figure 8. (b) Outcomes for cases d and e as a function of >i>v and collision angle θ. At each set of the initial conditions, all 100 runs all gave the same outcome.



Figure 9. (a) Total number of h-bonds nHB as a function of time shown for each initial droplet speed v. 57x42mm (300 x 300 DPI)


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Figure 9.(b) instantaneous temperature as a function of time shown for each initial droplet speed, v. The instantaneous temperatures at t = 0 for v = 790, 1000, 1300 m/s are not shown; they were 530, 640, and 910 K, respectively. 58x43mm (300 x 300 DPI)



Figure 10. Mixing progress expressed in terms of the m(τ) functions. If a m(τ) function stabilizes at 1, it is an indication that mixing is complete in that dimension(s). The initial conditions are shown at the top of each plot. In plots i to iii, since the y and z directions are identical, the results were obtained by averaging the two directions. Plots i to iv share the same legends, but plot iv has the additional legend for mz. 136x116mm (300 x 300 DPI)


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Table of Contents Graphic 44x24mm (300 x 300 DPI)


Free Energy and Dynamics of Water Droplet Coalescence - The

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