Wetting at the nanoscale: A molecular dynamics studyMohammad Khalkhali, Nasser Kazemi, Hao Zhang, and Qingxia Liu

Citation: The Journal of Chemical Physics 146, 114704 (2017); doi: 10.1063/1.4978497View online: http://dx.doi.org/10.1063/1.4978497View Table of Contents: http://aip.scitation.org/toc/jcp/146/11Published by the American Institute of Physics

THE JOURNAL OF CHEMICAL PHYSICS 146, 114704 (2017)

Wetting at the nanoscale: A molecular dynamics studyMohammad Khalkhali,1 Nasser Kazemi,2 Hao Zhang,1,a) and Qingxia Liu1,b)1Department of Chemical and Materials Engineering, University of Alberta, Edmonton,Alberta T6G 1H9, Canada2Department of Physics, University of Alberta, Edmonton, Alberta T6G 2E1, Canada

(Received 17 October 2016; accepted 1 March 2017; published online 21 March 2017)

A novel method to calculate the solid-liquid contact angle is introduced in this study. Using the 3Dconfiguration of a liquid droplet on a solid surface, this method calculates the contact angle along thecontact line and provides an angular distribution. Although this method uses the 3D configuration ofliquid droplets, it does not require the calculation of the 3D density profile to identify the boundaries ofthe droplet. This decreases the computational cost of the contact angle calculation greatly. Moreover,no presumption about the shape of the liquid droplet is needed when using the method introduced in thisstudy. Using this method, the relationship between the size and the contact angle of water nano-dropletson a graphite substrate was studied. It is shown that the contact angle generally decreases by increasingthe size of the nano-droplet. The microscopic contact angle of 83.0◦ was obtained for water on graphitewhich is in a good agreement with previous experimental and numerical studies. Neglecting othernanoscale effects which may influence the contact angle, the line tension of SPC/E (extended simplepoint charge model) water was calculated to be 3.6 × 10−11 N, which is also in good agreement withthe previously calculated values. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4978497]

I. INTRODUCTION

Wettability is a fundamental property of liquids govern-ing many environmental and industrial applications around us.This phenomenon is primary quantified by measuring the con-tact angle at the solid-liquid interface. When a liquid droplet isplaced on a rigid solid surface, the contact angle presents thebalance between three interfacial tensions which can be welldescribed by Young’s equation,1

cos(θ) =γs3 − γsl

γl3, (1)

where γsl, γs3 , and γl3 are the solid-liquid, solid-vapor, andliquid-vapor surface tensions. Once the liquid droplet touchesthe solid surface, the value of the contact angle decreases from180◦ to an equilibrium value, θ. When θ is smaller than 90◦ thesurface is considered to be hydrophilic, and when θ is largerthan 90◦ the surface is considered to be hydrophobic.

Despite the simplicity of wettability problem, the levelof uncertainty in the experimentally and/or theoretically mea-sured values of contact angles is fascinating.2 One source ofthe uncertainty in experiments comes from the fact that itis almost impossible to observe a unique contact angle fora given system. The contact angle values are often reportedas a range bounded dynamically by advancing and recedingangles.3 In this context, contact angle hysteresis which refersto the difference between advancing and receding angles is animportant parameter determining the resistance of a dropletupon sliding across a surface.4 Other sources of discrepanciesin the experimental results can be heterogeneity and/or impu-rity at the surface or in the liquid,2 sample preparation and

a)Electronic mail: hao.zhang@ualberta.cab)Electronic mail: qingxia.liu@ualberta.ca

measurement difficulties especially at the nanoscale,5 and pos-sible size effect.6 The latter is especially important at thenanoscale when the line tension (τ), the excess free energy perunit length of vapor-liquid-solid contact line, becomes signif-icant.7 The size dependent contact angle at the nanoscale isusually described with the “modified Young equation”,8,9

cos(θ) =γs3 − γsl

γl3−

τ

rγl3= cos(θ∞) −

τ

rγl3, (2)

where τ is the line tension, r is the radius of the contact line (itis assumed to be circular), and θ∞ stands for the contact anglesat r → ∞ (macroscopic contact angle which is calculated fromEquation (1)). Theoretically, the line tension is estimated to bebetween 10�11 and 10�10 N.10,11 The length scale at which theeffect of contact line tension is considerable can be estimatedby relating a typical value of line tension (τ ≈ 10−11−10−10 N)and surface tension (γlv ≈ 10−2 N/m) of water, which yieldsto τ/γlv in the order of nanometers.12 This means that highresolution characterizing techniques such as scanning elec-tron microscopy (SEM) and atomic force microscopy (AFM)are needed to capture the size dependence of the contactangle experimentally. There have been a number of experi-mental studies that have tried to approach the size dependentcontact angle at the nanoscale through such techniques.12–16

However, as explained thoroughly by Schimmele and Diet-rich,7,17 results of such studies should be interpreted withcare as many parameters may affect line tensions calculatedexperimentally.

Molecular modeling techniques are suitable tools that cancomplement experiments and provide deeper understandingespecially when studying wetting properties at the nanoscale.There have been numerous studies trying to address the wettingproblem using the molecular dynamics (MD) technique; how-ever, simulations are not free from uncertainties either. There

0021-9606/2017/146(11)/114704/12/$30.00 146, 114704-1 Published by AIP Publishing.

114704-2 Khalkhali et al. J. Chem. Phys. 146, 114704 (2017)

are two main approaches to calculate contact angles using MDwhich can be categorized as direct and indirect methods.4 Theindirect approaches try to evaluate the contact angle throughimplementing the calculated interfacial energies into Young’sequation (Equation (1)). In addition to the possible size effectmentioned above, the indirect method is unable to provideinformation about dynamics of liquid-surface interactions dueto its thermodynamic approach. Moreover, the correctness ofthis method is more questionable when it comes to study het-erogeneous and rough surfaces.4 Gao and McCarthy arguedthat the contact lines instead of the contact areas (and their cor-responding surface energies) are important in determining thesolid-liquid contact angle.18 These issues promote the directmethods which try to simulate the actual droplet with a finitesize on a solid surface. The size and time limit of MD simu-lations, however, give rise to some uncertainties regarding thesimulation results. A small droplet size may lead to a largecurvature and a large surface tension. The influence of dropletsize on the surface tension can be measured by consideringthe Tolman length which gives the extent by which the sur-face tension of a small liquid droplet deviates from its planarvalue,19

γ = γ0

(1 −

2δR

), (3)

where γ is the surface tension of droplet, γ0 is the surface ten-sion of liquid in its planar limit, δ is the Tolman length, and Ris the radius of the droplet. In the derivation of Equation (3),Tolman neglected the higher order terms and treated δ as aconstant. However, the necessity of higher order terms in pre-dicting the surface tension through Tolman treatment has beendiscussed excessively in the literature.20 Using the non-localmean-field density functional theory (DFT) for Lennard-Jones(LJ) fluid, Malijevsky and Jackson showed that the Tolmanapproach is valid for droplets with radii bigger than 10 diame-ters (10σ) while for smaller droplets a higher order curvaturedependence of the 1/R3 form is required in the Tolman treat-ment of the surface tension.20 They also estimated the Tolmanlength to be about a tenth of the molecular diameter and to havea negative sign. For the SPC/E water model, this would resultin δ ≈ −3 Å. There have been a number of other studies whichcalculated the Tolman length for water experimentally andtheoretically.21–23 Using MD simulations, this value is also cal-culated for different common water models.24,25 These studiesgive the Tolman length in the order of �0.5 Å. This means thatfor nano-droplets with the radii range between 1 and 10 nmthe difference between surface tension of nano-droplets and theflat surface is changing from 1% to 10%. Even if we ignore theTolman effect, the line tension may still cause size dependencyin nanodroplets as explained before. Malani et al. tried to solvethe droplet size problem by reversing the role of curvature ofthe liquid and solid.4 In their approach, a solid cylinder is sim-ulated in contact with liquid. Using the molecular dynamicsmethod, Seveno et al. also calculated the contact angle usingthe force distribution along a fiber dipped into a liquid.26 Thissimulation yielded a very good agreement with the AFM mea-sured force and the one predicted by Young’s equation. Themain drawback of these approaches is that the structure ofcurved surfaces of nanosolids is shown to be considerably

different than flat surfaces. It has been shown that nanopar-ticles can go through massive surface relaxations, sometimes,can even change their crystal structures.27,28 Moreover, thesemethods cannot provide information about the effect of theline tension when a possible size effect at the nanoscale is ofinterest.

Beside the time and size limits, calculating the con-tact angle from a MD simulation also accompanies sometechnical difficulties. One factor which makes contact anglecalculation from MD trajectories complicated is the iden-tification of atoms (or molecules) in a liquid droplet, gasphase, and at the liquid-gas interface. In the previous stud-ies, this was usually achieved through calculating the localnumber density.2,3,29–34 Calculating the 3D density profilefor small systems may be trivial but it becomes more timeand computational power consuming as the size of the sys-tem increases. After identifying the molecules in each phaseand at the liquid-vapor interface, the next challenge is tocalculate the contact angle. It is a usual practice to con-sider a partial spherical shape for the liquid droplet, such asthe one shown in Figure 1, and calculate the contact angleaccordingly.

Knowing the height, h, and the radius, r, of the partialsphere, the contact angle, θ, can be defined as

cos(θ) = 1 −hr

. (4)

Using Equation (4) and considering a constant densityin the liquid droplet, Hautman and Klein35 derived a rela-tion between the centre of mass of the droplet and the con-tact angle (see the appendix of Ref. 36 for details of thederivation),

〈zcom〉 = 2−4/3R0

(1 − cos(θ)2 + cos(θ)

)1/3 3 + cos(θ)2 + cos(θ)

, (5)

where zcom is the z coordinate of the centre of mass of thedroplet relative to the solid surface and 〈. . .〉 denotes the timeaverage. R0 is the radius of the fitted partial sphere to the dropletcalculated as

R0 =3N

4πρ0, (6)

where N is the number of the molecules in the droplet and ρ0

is the number density of the bulk liquid (0.033 Å−3

for water).As mentioned above, Hautman and Klein’s main assumptionto derive Equation (5) was that the density within the dropletis constant. However, it is well known that the density of theliquid close to the liquid-gas or liquid-solid interfaces deviates

FIG. 1. Schematic representation of the partial spherical shape.

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from the density in the bulk.37 To avoid the inaccuracy causedby assumption of the constant density within the droplet, Fanand Cagin calculated the height and radius of the best fittedpartial sphere and evaluated the contact angle directly fromEquation (4).36 They used a fine 3D density profile to esti-mate the interfacial area and volume of the liquid dropletwhich leads to the calculation of the height and radius of thebest fitted partial sphere through the following geometricalrelations:

V =π

6h(3c2 + h2), (7a)

Sbase = πc2, (7b)

c =√

h(2r − h). (7c)

de Ruijter et al. proposed an alternative approach tofind the best fitted partial sphere which includes findingthe liquid-gas interface through the density profile of thedroplet.33 The density of a liquid containing a liquid-gasinterface is usually modeled by the following sigmoidalfunction:37

ρ(r) =12

(ρl + ρ3) −12

(ρl − ρ3)tanh

(2(r − r0)

d

), (8)

where ρ(r) is the density of the liquid, ρl and ρ3 are the bulkliquid and gas densities, respectively, r0 is the position of theGibbs equimolar dividing surface, and d is the thickness of theinterfacial layer. In Equation (8), r0 and d are fitting param-eters and r is in the direction perpendicular to the interface.Considering the azimuthal symmetry and using the radial den-sity profile, de Ruijter et al. calculated r0 for the liquid dropletat different heights from the solid substrate. The best circularfit to the calculated r0 values was then found and the con-tact angle was calculated accordingly. This method is the mostcommon procedure used to calculate the contact angle fromthe molecular dynamics simulation results. Despite its sim-plicity, this kind of approach may result in inconsistent results,especially for small droplets.2 This is mainly because of thelarge shape fluctuation of the liquid droplet during an MDsimulation which makes the azimuthal symmetry unreliable.Moreover, the average shape of the liquid droplet may devi-ate from the spherical shape when studying inhomogeneoussurfaces. In nature, there are many examples of the inhomoge-neous surfaces which exhibit directional wetting properties.38

Directional wetting properties have been also made artificiallythrough altering the physical39 or chemical40 structures of thesurfaces. This is obvious that the presumption of the partialspherical shape of the liquid droplet may lead to a consider-able error when studying wetting properties of such surfaces.Despite the efforts made to improve the fitting by applyingmore complex procedures,41,42 the accuracy of methods usingthe 2D projection of the droplet to calculate the contact angleis still challenging.

To solve these issues, Santiso et al. proposed a methodwhich does not use any initial assumption about the shape ofthe droplet but uses the complete three-dimensional structureof the droplet near the surface to estimate the contact angle.2

At the first step of this method, the liquid-vapor interface isidentified using a discretized density profile. In the next step,interface molecules that are within a given distance from the

solid surface (zmax) are picked up and marked as an interfacecontact layer. The local contact angle at the position of eachwater molecule i in the interface contact layer is then calcu-lated using the normal to the plane that best fits the molecule iand its neighbors which are defined by a cutoff distance (rc).They applied a coarse-grained approach in order to exploresystem sizes beyond the atomistic simulation limits. The workof Santiso et al. provides a practical approach to calculate thecontact angle from the 3D structure of a droplet. However, theaccuracy of this method, like other methods using interfacerecognition to calculate the contact angle, is highly dependenton the density profile used to identify the contact layer. Aspointed out in their study, using a fine 3D mesh would result inthe recognition of the interface layer with a higher resolution;nevertheless, a very fine mesh may incorrectly mark densityfluctuations within the droplet as a part of the interface.2 Inaddition to the density profile resolution, the accuracy of thismethod is also highly sensitive to two other factors, namely,zmax and rc.

In this study, we propose a method which makes useof the well established convex hull algorithm to calculatethe distribution of contact angles along the contact line.No prior assumption on the shape of the liquid droplet isrequired for this method, and it does not use the density pro-file which makes it more time and computational efficient.Moreover, the contact angle calculation method proposed inthis study is much less sensitive to the predefined param-eters due to the robustness of the convex hull algorithm.We compared the performance of this method with previ-ously proposed algorithms. Because of its easy-to-implementnature, we believe this method can provide a practical toolto study the wetting phenomena via molecular dynamicssimulations.

II. METHODOLOGYA. Molecular dynamics simulation

A graphite-water system was used to test our contact anglecalculation algorithm. Graphite was chosen because its wet-ting properties have been studied excessively by experimentaland numerical studies.5,29,34,43–45 The parameters of water-graphite interaction potential were taken from the work ofWerder et al.,34 and the SPC/E model with a cutoff distance of10 Å was used for water. Similar parameters were used beforefor MD simulation of the graphite-water system.5 20 Å cutoffdistance was used for a short-range interaction between carbonand oxygen. As suggested by Rafiee et al.,44 this long cutoffwas intentionally selected to avoid any artificial contact angletransition because of the cutoff value of solid-liquid interac-tion. Particle-particle particle-mesh (PPPM) solvers were usedto handle the electrostatic interactions, and all MD simulationswere performed within the LAMMPS software package.46 Theperiodic boundary condition was used in all directions. A vac-uum gap was introduced in the z direction to build graphiteas an infinite 2D slab. The SPC/E water was first equilibratedthrough an NPT simulation at 300 K and 0 atm. Water dropletswith different sizes were then made by cutting partial spheresfrom the equilibrated water structure as shown in Figure 1.The main reason of using partial spheres was to decrease the

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computational cost of MD simulations. For each droplet size,the initial contact angle can be controlled by changing theheight (h) of the droplet (Figure 1),

h = rsin(θ0 − 90). (9)

To consider the effect of the initial contact angle of waterdroplets on the final contact angle (after equilibration), threedifferent initial shapes corresponding to the initial contactangles of 90◦, 100◦, and 110◦ were considered for each dropletsize. In the next step, water droplets were put on the graphitesurface. It has been shown that wetting properties of graphiteis influenced by the thickness of the film (number of graphenelayers).44,45 To eliminate the effect of thickness on the wet-ting properties, graphite slabs were made out of at least 6graphene layers. The size of the simulation box was chosenbig enough to avoid interactions between water droplets andtheir periodic images. Seven water droplets whose radii rangedfrom 1 to 7 nm were studied through a two step MD simu-lation. In the first step, energy minimization was performedto cure any possible close-contact between water moleculesor between water molecules and the graphite surface. Then,water droplets were relaxed through a NVT simulation at300 K for at least 5 ns. Simulation snapshots were savedevery 1 ps. The number of water molecules and size of thesimulation box for each model system are summarized inTable I.

In all simulations, carbon atoms were fixed in their crys-tallographic positions. This is acceptable as it has been shownthat fixing the carbon atoms would not affect the contact angle

TABLE I. Droplet and simulation box size for each system. r is the dropletradius, L is the parameter of the partial sphere according to Figure 1, θ0 is theinitial contact angle, and NW is the number of water molecules.

r (Å) L (Å) θ0 (◦) NW Box (Å×Å×Å)

3.420 110 9710 1.736 100 81 123.000 × 119.304 × 170.000

0 90 67

6.840 110 80720 3.473 100 669 123.000 × 119.304 × 170.000

0 90 528

10.261 110 279130 5.209 100 2346 134.100 × 130.870 × 220.000

0 90 1858

13.681 110 666140 6.946 100 5597 172.200 × 170.434 × 220.000

0 90 4420

17.101 110 1304550 8.682 100 10950 172.200 × 170.434 × 220.000

0 90 8661

20.251 110 2247260 10.419 100 18959 201.720 × 200.260 × 220.000

0 90 15013

23.941 110 3588970 12.155 100 30126 223.860 × 228.600 × 220.000

0 90 23912

greatly.34 The Nose-Hoover thermostat and a time step of 1 fswere used in all simulations.

B. Identification of the liquid droplet

Although our contact angle calculation algorithm does notrequire identification of the solid-liquid interface, it is still nec-essary to distinguish water molecules inside the droplet fromthose in the gas phase. We avoid using the 3D density profileto identify the droplet which enhances the speed of calcula-tion greatly. Instead of defining the liquid-gas interface viadensity information, we applied a hit-and-count method. Thehit-and-count method is faster than a density approach andcan define the interface efficiently. In the traditional densityprofile calculation, the first step is to define a 3D mesh to binthe data. Then the density profile is calculated by counting thenumber of data points in each bin and converting the num-ber of atoms per volume to the density. Finally, the liquid-gasinterface can be defined via a threshold limit on the density.However, for huge datasets, 3D binning is a cumbersome tech-nique with a high computational cost. To alleviate this problem,we use a hit-and-count method on three dimensional data ina 1D fashion. In this approach, we define a 3D mesh first,however, instead of binning the data in 3D, we do the bin-ning in each direction separately and then count the numberof elements in each mesh. Assume that we start from the x-direction, in other words, we only read the x-coordinate ofdata points and apply the binning process on them. If thenumber of data points in the 1D bin in the x-direction isless than a predefined value, we simply remove those datapoints from the dataset. We repeat this process successivelyon other dimensions (i.e., y- and z-directions). At the end, theremaining data points define the compact structure of the liquiddroplet.

As shown in Figure 2(b), after applying the hit-and-count technique, the sparse data points in the gas phase havebeen successfully removed. However, some of the scatteredpoints near the boundary of the droplet might have remaineduntouched (red points). To remove these points from thedroplet, we need a more precise algorithm. Our near boundaryscatter point removal algorithm works as follows. First, wedefine the tightest spherical cap, with radius R, which includesall data points in the droplet. Second, all data points in thesubregion between two spheres with radii R − ∆R and R + ∆Rwill be examined and classified as sparse gas phase or dropletdata point. Our criterion for classification is the Euclideandistance between the points. For a data point in the afore-mentioned subregion, if at least one data point exists with adistance smaller or equal to the predefined minimum value,which is based on the nearest neighbor distance according tothe radial distribution function of water, that data point willbelong to the droplet. On the other hand, if we cannot findany data points with distances smaller or equal to the prede-fined minimum value, that data point will be classified as gasphase and will be removed from the set. Moreover, all datapoints outside the sphere with radius R + ∆R will be consid-ered as sparse gas phase and all data points inside the spherewith radius R − ∆R will be classified as droplet data point.Hence, our search region includes only a small portion of thedata points which makes our algorithm more computationally

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FIG. 2. Configuration of the 10 nm water droplet after applying different steps of the liquid droplet identification algorithm: (a) Original configuration, (b) afterapplying the hit-and-count algorithm, and (c) after applying the final fine precision step.

efficient than the 3D density profile approach. Moreover,for most of the cases, the hit-and-count method is suffi-cient to achieve a compact structure of the droplet andeven if near-droplet gas molecules remain, they will notimpose a considerable error in the contact angle calcu-lation. As a result, applying the secondary fine precisionalgorithm to remove the near-droplet gas molecules is intro-duced as an optional step in the algorithm. Figure 2 showsthe effect of applying each step in identifying the liquiddroplet.

It is worth to mention that during the final fine precisionstep of the droplet identification process we made an assump-tion that the droplet shape is close to the partial sphere. Asa result, this step may not be accurate if there is an extremedeviation from the partial spherical shape. Such an extremecondition can happen in the cases such as liquid droplet impacton the solid surface or high temperature wetting studies. Inthese cases, it may be necessary to use density profiles todistinguish liquid and vapor molecules.

C. Contact angle calculation algorithm

In this section, we will explain the methodology that weuse for calculating the solid-liquid contact angle. Understand-ing the topology of the droplet (i.e., the surface separating theliquid from the gas phase) is a critical step which then canbe carried out by some geometrical calculations resulting in

the contact angle. After applying the hit-and-count method,we removed the outliers (i.e., atoms in the gas phase) fromthe data set, hence, the new data set is more compact and thesurface of the remaining data points can be well defined by aconvex hull. The smallest convex set that contains the pointsis a convex hull of that data set. Several interesting algorithmsare developed for calculating the convex hull for two, three,or higher dimension data sets. In this work, we adopt the wellknown Quickhull approach.47 The Quickhull algorithm is fastand can cope with imprecision in the data points (round offerrors with floating-point arithmetic). The method owns its fastcalculation to the point selection strategy. In Quickhull insteadof choosing a random point for evaluation, the furthest point isselected. Selecting the furthest point of the set has some advan-tages over random point selection methods. It results in a muchfaster algorithm when the dataset has few non-extreme points.Moreover, it uses less memory compared to other methodswhich is an added bonus when dealing with huge datasets espe-cially in higher dimensions. For three dimensional data points,the final output of Quickhull is a combination of some facetsbuilt from a subset of data that contains all other data points.

Here we explain the algorithm step by step. The Quick-hull algorithm is a recursive technique, and it uses a divide andconquer strategy. Consider a data set with N data points whichdefine a set S. The algorithm first finds two left and right fur-thest points in the set, namely, P1 and P2 points, and includesthese points in the convex hull. Now, the segment P1-P2

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connecting P1 and P2 divides the rest of the points into twosubsetsS1 andS2.S1 contains all of the data points in the left ofthe segment P1-P2, and S2 contains all of the data points in theright of the segment P1-P2. Until this stage, we divided the setS into two subsetsS1 andS2 and the points P1 and P2 are addedto the convex hull. To further evaluate the rest of the points, welook into the data points in each subset separately. To do so, weselect the furthest point in each subset from the segment P1-P2.Let us say in subset S1 the point P3 is the furthest point fromthe segment P1-P2. The algorithm will add the P3 point to theconvex hull. Now, the three segments connecting points P1, P2,and P3 define a triangle. Left, right, and inside of the triangledivide the subset S1 into three subsets S1 − left, S1-inside, andS1-right. In this step, if there are any points in the S1-insideset, those points will be removed from the set and the other twosubsets S1 − left and S1 − right will be evaluated further withthe segments P1-P3 and P2-P3, respectively. The same pro-cedure would be followed for the S2 subset. This divide andconquer process will be continued until there are no pointsinside the subsets. Finally, the points included in the convexset will construct the convex hull of data points of the set S.The output is a combination of thick facets that contain all pos-sible exact convex hulls of the input. In our simulations, whichis in 3D, the facets are triangles. Moreover, we will remove thefacets that do not contribute to the volume of the convex hull.This will result in a topology that has a more concise form(Figure 3).

The topology and structure of the convex hull will help usto define the distribution of the solid-liquid contact angle. Aswe mentioned earlier, the convex hull is a combination of sometriangles with different areas. We can calculate the angle ofthe convex hull in the location of these triangles. Each triangleis built via three points from the set S. These points definethree vectors from the origin as v1, v2, and v3. One can easilycalculate the normal of the triangle as

n = (v1 − v2) × (v1 − v3), (10)

where × denotes the cross product, n = n1i + n2j + n3k is thenormal of the triangle, and i, j, and k are the unit vectors inthe x, y, and z directions, respectively. Vectors of each triangleare sorted in a way that n is pointing outwards from the tri-angle. Now the angle of the triangle with respect to the solid

FIG. 3. A convex hull of the 10 nm water droplet built via the Quickhullalgorithm.

FIG. 4. An example of angular probability distribution calculated from thecontact angle calculation method in this study.

surface is

θ = cos−1 *..,

n3√(n2

1 + n22 + n2

3)

+//-

. (11)

We will repeat the process for all of the triangles in theconvex hull which are close to the solid surface. This meansthat to calculate the contact angle with the solid surface wewill only consider those triangles that contain points smallerthan the predefined limit in the z direction. At this point, wecalculate the histogram bin counts for all calculated angles.When working with floating numbers, the algorithm repro-duces a big number of triangles with smaller areas. To remedythis difficulty and remove erroneous triangles (i.e., adjacentnon-convex facets), the Quickhull algorithm will merge oneof the facets into a neighbour in a way that the merging pro-cess minimizes the maximum distance of the vertex to theneighbour.47 However, it is still possible to get some triangleswith smaller areas that do not give proper information aboutthe shape of the droplet. To make our algorithm more robust,we use weighted histogram bin counts. In the weighted his-togram, each calculated angle is weighted by the area of thecorresponding triangle. Lastly, we repeat the same process forN number of simulation snapshots and report the probabilitydistribution of all calculated angles. One may define the anglewith the maximum number of counts as the solid-liquid contactangle for the simulation points in the set S. However, repre-senting the contact angle by the distribution provides moreinformation about the wetting behavior of the liquid. Figure 4shows an example of angular probability distribution calcu-lated from our contact angle calculation code. Previous con-tact angle calculation algorithms described in Section I werealso coded, and the corresponding codes are provided in thesupplementary material along with the codes for our method.All contact angle calculations were performed on the last500 ps of simulations.

III. RESULTS AND DISCUSSION

Before getting into the details of the results of our contactangle calculation method, we provide a thorough comparison

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between this method and other available contact angle calcu-lation methods in the literature. In order to perform such acomparison successfully, we should take the following pointsinto consideration:

• In the previous calculation methods, the contact anglewas always represented as the average value. As men-tioned before, our method reports the distribution ofthe calculated contact angles. Thus, such a distri-bution was also calculated for other methods to beable to make a proper comparison. This means thatthe contact angles were calculated for each atomicconfiguration and then represented as a distribution.Methods proposed by Hautman and Klein,35 Fanand Cagin,36 and de Ruijter et al.33 produce a sin-gle contact angle for each atomic configuration butthe method of Santiso et al.2 and the method pro-posed in this study calculate multiple contact anglesfrom the 3D structure of the droplet at each atomicconfiguration.• For methods proposed by Hautman and Klein35 and Fan

and Cagin,36 a prior process on droplets are requiredto identify whether the molecules belong to the liquidphase. As such a process was not explained in the orig-inal papers, we used our liquid droplet identificationalgorithm before applying these methods.• The accuracy of the methods proposed by de Ruijter

et al.33 and Santiso et al.2 highly depends on the reso-lution of the density profile. Dimensions of the 3D meshused to calculate the density profile for these meth-ods were tuned to achieve the best combination of thecalculation accuracy and the computational time.

Figure 5 shows the comparison between the computa-tional time of different contact angle calculation methods. Allalgorithms were coded in MATLAB Version 8.6.0.267264 andcalculations were performed using a computer with two Intel

© Xeon r E5-2620 @ 2.0 GHz CPUs and 62.7 GB of RAM.Droplets with the initial contact angles (θ0) of 90 Å wereused.

As expected, Figure 5 shows that the methods requiringdensity profile calculations are much more time consumingthan others and their computational cost increases exponen-tially as the size of the droplet increases. The method of Santisoet al.2 shows the highest computational cost due to its extensive3D density profile calculation and the plane fitting procedure.It is apparent that the method proposed in this study is com-putationally efficient with the computational cost comparablewith methods proposed by Hautman and Klein35 which simplyuses the centre of mass of the droplet to calculate the contactangle. It should be pointed out that the fine 3D mesh (0.4 Å)used by Fan and Cagin36 to calculate the volume of the liquiddroplet and the interfacial area is not practical for the size rangethat we studied in this work. Thus, we used the convex hullalgorithm to calculate the area and volume of droplets whichdecreased the computational cost of this method greatly. Thecontact angle was then calculated through applying the vol-ume and area obtained from the convex hull triangulation intoEquations (4) and (7). It is also worth to mention that the cal-culation time for our method was measured while the fine pre-cision droplet identification step was applied. As explained inSection II, the hit-and-count procedure is often sufficient andthe removal of the fine precision droplet identification stepwould improve the computational cost of our method evenmore. The contact angle distributions calculated from eachalgorithm are compared in Figure 6. As mentioned before, themethod of Santiso et al.2 is highly sensitive to its predefinedparameters. Two different contact angle distributions repre-sented for this method in Figure 6 were obtained using twodifferent sets of parameters, i.e., those given in the originalpaper (zmax = 0.5 nm and rc = 1 nm) and those tuned for a10 nm droplet in this study (zmax = 0.75 nm, rc = 0.5 nm). Boxplots are also used to represent the calculated contact angle

FIG. 5. Comparison between the calculation time of dif-ferent contact angle calculation methods. Droplets withthe initial contact angles of (θ0) of 90◦ were used. All cal-culations were performed on the atomic configurations ofthe last 500 ps of 5 ns simulations which were saved at1 ps intervals. The inset plot shows the calculation timefor the method proposed by Santiso et al.2

114704-8 Khalkhali et al. J. Chem. Phys. 146, 114704 (2017)

FIG. 6. Contact angle probability distribution graphs calculated for the 10 nm droplet using different algorithms. (a) shows the probability distribution of thecontact angle during the last 500 ps of the 5 ns simulation, and (b) represents the same distribution using box plots. Box plots depicted in this study representdistributions as follows: x, error bars, the upper and lower box limits, the band inside the box, the square inside the box, and the hollow circle represent theminimum and maximum values, the 5th and 95th percentiles, the 25th and 75th percentiles, the median, the mean, and the mode, respectively.

distributions as they can provide the important informationabout the distributions concisely.

Figure 6 shows that the three methods which make useof the spherical shape assumption result in a narrower contactangle distribution. It is expected as the fitting to a predefinedshape would neglect the deviation of the shape of the liquiddroplet from the partial spherical shape. This deviation is moresignificant in smaller droplets because of the large shape fluc-tuation during the MD simulation. As mentioned before, thepresumption of the partial spherical shape of liquid dropletsmay also cause major inaccuracies when studying the wettingproperties of the inhomogeneous surfaces as the shape of thedroplet on these surfaces may deviate from the spherical shapegreatly. While all of the first three methods make the presump-tion of the spherical shape of the liquid droplet, it seems that themethod proposed by de Ruijter et al.,33 which finds the fittedpartial sphere equation through a radial density profile, resultsin a slightly bigger contact angle value than the other two. Theaverage contact angle calculated by this method is also veryclose to the one calculated with our method. This is becausefor the system used to build Figure 6, which includes a per-fect solid surface without any defects and a fairly large liquiddroplet, the partial spherical presumption does not impose asignificant inaccuracy. The method proposed by Santiso et al.2

does not make any presumption about the shape of the droplet;nevertheless, it is very sensitive to its predefined values. Apply-ing the parameters used by the authors, we calculated a similarcontact angle distribution as reported by them.2 By fine tun-ing the predefined parameters, however, we could achieve abetter contact angle distribution which is in the range of thosecalculated by de Ruijter et al.33 and our methods. The resultsof Figures 5 and 6 show that the contact angle calculationmethod developed in this study provides the best combinationbetween the computational cost and calculation accuracy, yetit does not make any presumption about the shape of the liquiddroplet. This method was used to calculate the contact angle

of water on graphite for all systems listed in Table I, and theresults are represented in Figure 7. These distributions werecalculated using the atomic configurations of the last 500 psof 5 ns simulations.

Figure 7 clearly shows that increasing the size of thedroplet results in narrower and more uniform contact angledistributions. The large variation of contact angles in the smalldroplets can be a combination of the results of multiple systemproperties. First of all, the convex hull triangulation results ina less number of triangles when the number of points in thedata set decreases. In other words, it is more difficult to definean accurate representation of the surface of the liquid dropletthrough convex hull triangulation for the small droplets. Forinstance, the smallest droplet (R = 1 nm) is made of lessthan 100 water molecules which is more accurate to call ita water cluster. Moreover, small droplets suffer from largershape fluctuations during MD which is probably the main rea-son broadening the contact angle distribution as the size ofthe droplet decreases. The shape fluctuation of water dropletsduring an MD simulation can be quantified through geomet-ric parameters such as the sphericity. The sphericity, Ψ, of ageometric object is defined as the ratio of the surface area of asphere with the same volume as the given object to the surfacearea of the object,

ψ =

3√π(6V )2

A, (12)

where V and A are the volume and area of the object, respec-tively. We can calculate the sphericity of water droplets dur-ing MD simulations through the volume and surface valuesobtained from the convex hull triangulation. The results ofthese calculations are shown in Figure 8.

Figure 8 clearly shows that the fluctuation in the shapeof droplets decreases by increasing the size. According toFigure 8, it can be concluded that the broad contact angledistribution of the smallest droplet (Figure 4) resulted from its

114704-9 Khalkhali et al. J. Chem. Phys. 146, 114704 (2017)

FIG. 7. Contact angle probability distribution graphs calculated for droplets with different sizes and initial contact angles: (a) R = 1 nm, (b) R = 2 nm,(c) R = 3 nm, (d) R = 4 nm, (e) R = 5 nm, (f) R = 6 nm, and (g) R = 7 nm. Snapshots of the last 500 ps of 5 ns simulations were used to construct probabilitydistribution graphs.

large shape fluctuation. This broad contact angle distributionincreases the uncertainty in the calculated contact angle values.Although using big droplets would result in a more accuratecontact angle calculation, the computational cost, as anotherfactor, limits the size of the water droplet in a MD simula-tion. As the size of the water droplet increases, the number ofwater molecules increases proportional to R3 (R is the radiusof the droplet). In the droplet size range that examined in ourstudy, it seems that a 10 nm (R = 5 nm) droplet offers the bestcompromise between the contact angle calculation accuracyand computational demand. However, as mentioned before,the contact angle at the nanoscale should be treated with caredue to the possible size dependence. To show the effect ofthe size on the contact angle, equilibrium contact angles of

droplets with different sizes but the same initial contact angles(θ0 = 110◦) are compared in Figure 9.

A general trend of decreasing the average contact angle byincreasing the size can be seen in Figure 9. Although a smallerdroplet size range was used, this trend has been also reported inprevious MD studies on the same system.34,48,49 These studiesattributed this trend to the effect of the line tension accordingto Equation (2). However, it should be considered that thereare multiple nanoscale effects which may influence the con-tact angle at the nanoscale. This causes a considerable amountof the uncertainty about the origin of the size-dependent wet-ting at the nanoscale. Let us consider the very small droplets(R< 3 nm) in which we have shown that the large shape fluc-tuation and the small number of the data points make the

114704-10 Khalkhali et al. J. Chem. Phys. 146, 114704 (2017)

FIG. 8. The sphericity distribution of water droplets with different sizes dur-ing the MD simulations. Since the sphericity of the smallest droplet (R = 1 nm)varies in a much bigger range compared to the others, different axes are usedfor this droplet (left axes).

contact angle hard to define and calculate accurately. It wasalso emphasized by Sergi et al. that the contact angle can nolonger be derived accurately from the tangent to a circular fitwhen the size of the droplet is smaller than 3 nm.29 As men-tioned before, the Tolman treatment of the liquid-vapor surfacetension is not valid in this size range either. A large devia-tion of the liquid-vapor surface tension from the bulk valuemay also affect the contact angle. It is also a well establishedfact that there is a large fluctuation in the liquid density nearthe solid surface due to the layering of liquid phase. Beckeret al. argued that this effect may cause a decrease in the liquid-vapor and solid-liquid interfacial tensions.50 This is especiallyimportant in the very small droplets where the lack of the bulkliquid properties is significant. Scocchi et al. also reported adeviation from the modified Young’s equation (Equation (2))for droplets smaller than 3 nm.29,51 They proposed that thisunexpected behavior is due to the fact that molecules nearthe contact line experience reduced cohesive forces and thisresults in an increase in the adhesive component of the total

FIG. 9. Equilibrium contact angle distributions for droplets with differentsizes calculated from the last 500 ps of the 5 ns simulations.

force which tends to increase the contact area.51 Although ourcontact angle calculation method shows the increase in contactangle value by decreasing the size even for droplets smallerthan 3 nm, the contact angle values calculated for the smalldroplets should still be handled with care.

If we do not consider the very small nanodroplets andneglect the effect of other mentioned nanoscale effects, theline tension may be calculated from the size-dependent contactangle (Equation (2)). It is necessary to define the contact angleas a single value to be able to calculate the line tension. It can beseen in Figure 7 that contact angle distributions are not normaland show a positive skewness. The median value of a distribu-tion is less affected by outliers and skewed data; however, themean would reflect the effect of asymmetry which is impor-tant in the case of contact angle distributions. Thus, the mean isprobably the best choice to represent the contact angle througha single value. Moreover, choosing the mean value would pro-vide a consistency with previous studies. In addition to thecontact angle value, we need the radius of the contact lineto be able to estimate the line tension through Equation (2).The interfacial area of the liquid droplet and the solid surfacewas calculated through convex hull triangulation. The radiusof the contact line then was estimated by a circular fitting tothe interfacial area. In Figure 10, the relationship between thereciprocal of the radius of the contact line and cosine of thecontact angle is shown.

Through Equation (2), the reciprocal of the contact lineradius and the contact angle has a linear relationship at whichthe intercept and slope correspond to the cosine of the macro-scopic contact angle (cos(θ∞)) and the line tension over thesurface tension (τ/γl3), respectively. Figure 10 clearly showsthat the contact angles calculated from the smallest droplets(R = 1 nm, θ0 = 90◦, 100◦, 110◦) are out of the linear regime.As explained thoroughly before, the accuracy of contact anglescalculated from such a small droplets is of a great doubt. θ∞calculated from the linear fit in Figure 10 is 83.0◦. This valueis slightly smaller than the one calculated by Werder et al.34

(θ∞ = 86◦) but it is in a very good agreement with the one

FIG. 10. The relationship between the reciprocal of the contact line radiusand the contact angle. The smallest droplets (R = 1 nm, θ0 = 90◦, 100◦, 110◦)were excluded from the linear fit.

114704-11 Khalkhali et al. J. Chem. Phys. 146, 114704 (2017)

calculated by Dutta et al. using the same potential param-eters (θ∞ = 83◦).48,49 This value is also in good agreementwith the experimentally measured contact angles of water ongraphite.44,45 Using the previously calculated surface tensionof the SPC/E water model (γl3 = 0.0636 N/m)52 and the slopeof the fitted line in Figure 10 (τ/γl3 = 0.56 nm), the line ten-sion of water on graphite is calculated to be 3.6×10−11 N. Thisvalue is also in a very good agreement with the range of waterline tension values predicted theoretically.10,11 It is also in goodagreement with previous numerical53 and experimental54 stud-ies which predicted the upper band of the line tension valueto be 5× 10−11 and 10�10 N, respectively. Moreover, there isa very good agreement with the line tension values calculatedvia molecular dynamics simulations of the graphite-water sys-tem using the same forcefield parameters34,48 (see corrigendaof these publications).

Here, it is worth to emphasize again that some assump-tions were made in the line tension calculation reported here:First, we neglected the other nanoscale effects which mayinfluence the solid-liquid and liquid-vapor interfacial tensions.Due to the weak dependency of the line tension to the sizeof the nanodroplets, we cannot conclude confidently that thesize-dependency of the contact angle is solely caused by theline tension effect. Second, the contact line curvature was esti-mated as the radius of a circle with the same area as thesolid-liquid interface. This assumption implies that the linetension is homogeneous along the contact line; however, itis well known that the line tension is dependent on the localcurvature of the contact line.

IV. CONCLUSION

A novel method to calculate the solid-liquid contact anglefrom molecular dynamics simulations was introduced. Thismethod does not make any presumption about the shape of theliquid droplet. Although the 3D configuration of the dropletis used, this method does not require the calculation of the3D density profile to identify the liquid droplet boundaries.This decreases the computational cost greatly and improvesthe speed of the calculation especially for big systems. Afteridentifying the droplet, the code makes use of the well knownQuickhull algorithm to calculate the convex hull of the liquiddroplets. Distribution of contact angles along the contact lineis then calculated through the convex hull triangulation. Usingthis method, we studied the size dependence of the contactangle of water nanodroplets on the graphite substrate. It wasshown that for the water droplets with radii ranging from 1to 7 nm, the contact angle generally decreases by increasingthe size. Neglecting the other nanoscale effects which maycause this size-dependent wetting behaviour, the line tensionof SPC/E water on graphite is calculated to be 3.6 × 10−11 Nwhich is in a good agreement with previously reported values.We believe that the contact angle calculation method intro-duced in this study can be helpful for future studies on wettingproperties via molecular dynamics simulations. One possiblepath for the future research in this area is to study the wet-ting properties of non-ideal surfaces. Having a reliable contactangle calculation algorithm which is capable of capturing con-tact line inhomogeneities, the relationship between the contact

line fluctuation and contact line pinning can be explored forthe non-ideal surfaces.

SUPPLEMENTARY MATERIAL

See supplementary material for the contact angle cal-culation codes and sample molecular dynamics trajectoryfiles.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the support of theNatural Sciences and Engineering Research Council of Canada(NSERC) and the Canadian Centre for Clean Coal/Carbon andMineral Processing Technologies (C5MPT). Computationalresources are provided by Westgrid.

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