Wettability of graphitic-carbon and silicon surfaces: MD modeling and theoreticalanalysisBladimir Ramos-Alvarado, Satish Kumar, and G. P. Peterson Citation: The Journal of Chemical Physics 143, 044703 (2015); doi: 10.1063/1.4927083 View online: http://dx.doi.org/10.1063/1.4927083 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/143/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Ice and water droplets on graphite: A comparison of quantum and classical simulations J. Chem. Phys. 141, 204701 (2014); 10.1063/1.4901562 Interfaces of propylene carbonate J. Chem. Phys. 138, 114708 (2013); 10.1063/1.4794792 Coarse-graining MARTINI model for molecular-dynamics simulations of the wetting properties of graphiticsurfaces with non-ionic, long-chain, and T-shaped surfactants J. Chem. Phys. 137, 094904 (2012); 10.1063/1.4747827 Anomalous wetting on a superhydrophobic graphite surface Appl. Phys. Lett. 100, 121601 (2012); 10.1063/1.3697831 Squeezing wetting and nonwetting liquids J. Chem. Phys. 120, 1997 (2004); 10.1063/1.1635813

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

128.61.141.172 On: Fri, 24 Jul 2015 14:14:41

THE JOURNAL OF CHEMICAL PHYSICS 143, 044703 (2015)

Wettability of graphitic-carbon and silicon surfaces: MD modelingand theoretical analysis

Bladimir Ramos-Alvarado, Satish Kumar, and G. P. Petersona)

The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta,Georgia 30332, USA

(Received 27 April 2015; accepted 9 July 2015; published online 24 July 2015)

The wettability of graphitic carbon and silicon surfaces was numerically and theoretically inves-tigated. A multi-response method has been developed for the analysis of conventional moleculardynamics (MD) simulations of droplets wettability. The contact angle and indicators of the qualityof the computations are tracked as a function of the data sets analyzed over time. This methodof analysis allows accurate calculations of the contact angle obtained from the MD simulations.Analytical models were also developed for the calculation of the work of adhesion using themean-field theory, accounting for the interfacial entropy changes. A calibration method is proposedto provide better predictions of the respective contact angles under different solid-liquid interactionpotentials. Estimations of the binding energy between a water monomer and graphite match thosepreviously reported. In addition, a breakdown in the relationship between the binding energy andthe contact angle was observed. The macroscopic contact angles obtained from the MD simulationswere found to match those predicted by the mean-field model for graphite under different wettabilityconditions, as well as the contact angles of Si(100) and Si(111) surfaces. Finally, an assessment ofthe effect of the Lennard-Jones cutoff radius was conducted to provide guidelines for future compar-isons between numerical simulations and analytical models of wettability. C 2015 AIP PublishingLLC. [http://dx.doi.org/10.1063/1.4927083]

I. INTRODUCTION

Carbon-based nanomaterials have been extensively inves-tigated from the electrical, mechanical, and thermal perspec-tives; on the other hand, the characterization of these mate-rials in aqueous environments has received relatively moderateattention; regardless of this, high quality investigations havebeen conducted on this topic. As early as 1940, Fowkes andHarkins1 measured graphite contact angles in the range of85.3◦–85.9◦. In 1970, Morcos2 reported a contact angle of84.2◦ for cleavage graphite. The vast majority of these con-tact angle measurements were made in samples exposed toair. Schrader3 conducted experiments in an ultrahigh vacuumenvironment and obtained a contact angle of 35◦ ± 4◦. Thislarge variation between the measured graphite contact anglesin air and vacuum was attributed to the organic contaminationof the samples.3 In more recent investigations, contact angles of98.3◦ ± 5.1◦4 and 91◦ ± 1◦5 were measured on highly orderedpyrolytic graphite (HOPG) surfaces exposed to air. Li et al.6

assessed the effects of airborne hydrocarbon contamination onthe wettability of graphene coated surfaces and HOPG. A con-tact angle of 64.4◦ was measured on a free-of-contaminantsgraphite surface. Although the contamination of the graphiticsurfaces had been reported by Schrader,3 the findings by Liet al.6 were somewhat unexpected and as a source of uncer-tainties in the review of the current investigations of the wettingtransparency of graphene.7

a)E-mail address: bud.peterson@gatech.edu

Molecular dynamics (MD) simulations have been used inprevious investigations to evaluate the wettability of varioussurfaces. A number of methods have been developed to extractthe macroscopic contact angle, θ∞, from the MD simulations.These include (1) a method in which the surface free energiesare obtained from statistical calculations of the stresses8 andalgorithms such as the phantom wall.9 Young’s equation is thenused to determine θ∞; (2) the microscopic contact angles anddroplet base radii are obtained from hemispherical droplets ofdifferent sizes, allowing θ∞ to be obtained through data fittingusing the modified Young’s equation; (3) cylindrical liquidslabs are equilibrated over atomically flat surfaces, and assum-ing that the contact line is straight, the microscopic contactangle is considered to be similar to its macroscopic counterpartas predicted by the modified Young’s equation.

Werder et al.10 conducted a parametric analysis of graphitewettability using MD simulations. A calibration of the oxygen-carbon Lennard-Jones (LJ) potential parameters (εCO andσCO)was conducted to obtain θ∞ = 86◦ using the hemisphericaldroplet method. An oxygen-carbon potential cutoff radius, rc,of 10 Å and a two graphene layer substrate were sufficientto obtain consistent values of θ∞. In the calibration process,Werder et al.10 observed a linear relationship between εCO andthe calculated contact angles. Moreover, a more fundamentallinear relationship was found to exist between the bindingenergy (Eb) and the contact angle. Jaffe et al.11 attempted toverify the results of the previous investigation. The effect of rcon the contact angle calculations, including hydrogen-carboninteractions, were reevaluated and appeared to justify the use ofonly two graphene layers to model the bulk graphite, based on

0021-9606/2015/143(4)/044703/11/$30.00 143, 044703-1 © 2015 AIP Publishing LLC

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

128.61.141.172 On: Fri, 24 Jul 2015 14:14:41

044703-2 Ramos-Alvarado, Kumar, and Peterson J. Chem. Phys. 143, 044703 (2015)

an Eb analysis. It was concluded that the previous recommen-dation for rc was not large enough to produce rc—independentcontact angles. The investigation of Walther et al.12 concludeda series of three articles published by the same group thataddressed the wettability of carbon surfaces. In the last of thethree publications, the authors applied the previously refinedmethods to account for impurities in the water as well as inthe solid. References 10–12 are commonly taken as a startingpoint for any MD wettability analysis.

Scocchi et al.13 performed a comparative analysis ofgraphite wettability using spherical and cylindrical dropletsand proposed new LJ potential parameters to obtain an exper-imental value of θ∞ = 127◦. Different wettability regimes fordroplets of different sizes as well as a marked size effect incylindrical droplet simulations were observed, when thereshould be none or minimum size effects.14–16 Sergi et al.17

observed deviations from the modified Young’s equation forsmaller droplets and a sudden transition from size-affectedbehavior to bulk calculations of contact angles from the MDsimulations. In a more fundamental investigation, Wu andAluru18 used ab initio calculations to obtain the non-bondedinteraction potentials between graphitic carbon and water. Theresulting potentials were fitted to be applied in MD simulationsand underpredictions of the contact angle were found whenusing these potentials compared to the calibrated ones. Itwas also observed that molecular models with similar Eb ledto different contact angle calculations and hypothesized thatEb was not as important as previously suggested. Driskillet al.19 reported a good agreement between the values ofthe binding and zero potential distances for graphene andwater obtained via ab initio simulations and using empiricalmodels; however, a wide range of binding energies were foundfrom quantum calculations over the literature. Therefore, theempirical parameterization of LJ potentials was suggested asan alternative to ab initio derived potentials due to the differentbinding energies predicted and the adjustments necessary to beapplied in a pairwise additive fashion.

Unlike the wetting characterization of graphitic-carbonsurfaces, silicon has not been extensively investigated pri-marily because of the rapid natural growth of a SiO2 layerwhen silicon is exposed to air. Barisik and Beskok20 widelyacknowledged this issue and reported a comprehensive reviewof the experimental investigations on the wettability of silicon.The contact angle of silicon has been reported to vary from35◦ to 96◦, but recent measurements on clean surfaces havenarrowed down this range to somewhere between 86◦ and89◦. As a first estimate, Barisik and Beskok20 assumed thecontact angle of Si(100) to be 88◦ and found the correspond-ing values of εSi–O and σSi–O to obtain this angle from MDsimulations.

Mean-field (MF) theory has been successfully used todevelop analytical models of wettability.19,21,22 The MF theoryessentially reduces the complexity of a many-body problemto an approximated single-particle averaged one. The single-particle treatment of the wettability problem depends on anappropriate description of the density distribution of particlesat the interface and on a single-particle potential. In most MFmodels of wettability, the work of adhesion only considersthe solid-liquid interaction energy and neglects the interfacial

entropy contribution. Recently, Taherian et al.23 determinedthat the interfacial entropy accounts for∼30% of the total workof adhesion in graphite. Taherian et al.24 later developed ananalytical model based on free energy perturbation theory toobtain the interfacial entropy. We propose a calibration methodfor a MF-based model of wettability to solve the problem of theoverprediction of density using Boltzmann distributed parti-cles. The calibration process is performed on the descriptionof the water density distribution function for its utilization ina single-particle potential in order to match a given θ∞. Weconducted a comparison between our model, a sharp-kink-approximation (SKA)-based model, and data extracted fromMD simulations. Not only accurate analytical predictions of θ∞but also other parameters obtained via MD were obtained for awide range of surface affinities. Additionally, the capabilitiesof such model to predict Eb and the contact angles as a functionof rc were investigated. We corroborated the fact that there is astrong correlation between Eb and the contact angle. Further-more, we found a breakdown in this relationship by simplychanging the interaction potential. On the MD modeling side,guidelines are proposed for conducting a thorough analysis ofthe data obtained from MD simulations of wettability duringpostprocessing. Accurate and reliable calculations are obtainedby following the process suggested here for graphite and sili-con surfaces in different crystallographic planes.

II. ANALYTICAL MODEL OF WETTABILITY INCLUDINGINTERFACIAL ENTROPY

The analytical formulation of wettability begins fromassuming that water and a solid substrate interact through aLJ potential given by

Vij (r) = 4εij

(σij

r

)12−

(σij

r

)6, (1)

where r is the distance between two particles, εij and σij are theenergy and distance parameters, respectively, and i j representa pair of interacting atoms. Within the framework of the MDmodeling of water wettability, εij and σij are the LJ parametersof the interaction between oxygen and the solid atoms sincehydrogen-solid interactions are commonly neglected. If thesolid substrate is assumed to infinitely extend in the x-y planewith a constant atomic density per unit area ρs, the interactionpotential between a water molecule and the substrate (w(z))can be obtained from

w (z) = ρs

S

Vijdxdy = ρs

rlim0

Vij

(z2 + r ′2

)2πr ′dr ′, (2)

where S represents the surface generated when the interac-tion zone of a single particle (spherical space) is truncatedby a solid substrate and r ′ = x2 + y2, z is assumed constant,and rlim =

r2c − z2 is the integration limit of r ′. Integrating

yields

w (z) = 4πρsεσ2(

σ10

5z10 −σ4

2z4

)−

(σ10

5r10c− σ4

2r4c

), (3)

where w(z) is defined in z ∈ [0,rc]. For the particular case ofgraphitic-carbon, Eq. (3) represents the interaction potential

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

128.61.141.172 On: Fri, 24 Jul 2015 14:14:41

044703-3 Ramos-Alvarado, Kumar, and Peterson J. Chem. Phys. 143, 044703 (2015)

between one graphene layer and a single water molecule. If it isassumed that van der Waals (vdW) interactions are additive, thesingle-particle potential for graphite (wN(z)) can be obtainedby adding a number N of w(z) graphene potentials. This

can be mathematically achieved by shifting w(z) a discretedistance kh0, where h0 is the interlayer distance betweengraphene layers and k = {0,1,2, . . . ,N − 1}. This is illustratedin

wN (z) = 4πρsεσ2N−1k=0

σ10

5(z + kh0)10 −σ4

2(z + kh0)4−

(σ10

5(rc + kh)10 −σ4

2(rc + kh)4)

. (4)

The function shift operation was performed in such a way thata direct application of Eq. (4) is possible without affecting z.Constants are not affected by shifting, but in this particularcase if rc is not shifted together with z, wN(z) would be evalu-ated outside of its domain causing divergence at small rc andlarge N . Eq. (4) was previously introduced for modeling theinteraction potential between graphite and a polymer atom byDaoulas et al.25 and originally by Steele26 to model graphite-gas interactions. The binding energy of the single-particlemodel can be obtained as Eb = min[wN(z)] and the total solid-liquid interaction energy per unit area (∆UWS/A) can be foundby integrating wN(z) over the density distribution of the liquidparticles in the z-direction as follows:

∆UWS/A = −rc

zref

ρL (z) wN (z) dz, (5)

where ρL(z) is the liquid particles density per unit volume andzref can be defined as wN(zref) = 0. Taherian et al.24 applied thefree energy perturbation theory to obtain an expression for thework of adhesion,

Wa =1A(∆UWS − T∆SWS) = γlv (1 + cos θ) , (6)

where−∆SWS/A is the entropy loss due to the ordering imposedon the interfacial water molecules closed to the solid surface,γlv is the liquid-vapor surface tension, and θ is the contactangle. As indicated in Refs. 23 and 24, this is not the thermody-namic property ∆S where water self-interactions are includedand later canceled with the corresponding self-interaction en-ergy contribution from ∆U in the definition of Wa. The entropyterm in Eq. (6) is obtained from

∆SWS = NWkB ln

zmaxzmin

PA (z) exp [δwN (z) /kBT] dz

(7)

FIG. 1. (a) Si(100) and (b) Si(111) structures normal to the wetted plane. Thewetted surface is the x-y plane.

and

PA (z) = exp [−wN (z) /kBT]zmaxzmin

exp [−wN (z) /kBT] dz, (8)

where δwN (z) = wN (z) − ⟨wN (z)⟩A, the average ⟨wN (z)⟩A isevaluated assuming that wN (z) is Boltzmann distributed with

probability density PA(z), NW/A =zmaxzmin

ρL (z) dz, kB is the

Boltzmann constant, and T is the absolute temperature. Tahe-rian et al.24 recommended values of zmin = 0.22 nm and zmax= 0.88 nm to properly capture ∆SWS/A at the interface. Theyalso found that using a Boltzmann distribution27 to representρL(z) significantly overestimates NW/A compared with MDresults. The SKA showed a better accountability of NW/A buta substantial underestimation of ∆UWS/A, rendering imprac-tical a fully analytical calculation of Wa. We propose here touse ρL (z) = ρL,0 exp [−wN (z) /ηkBT], where ρL,0 is the bulkdensity of the liquid particles and η is the only tuning parameterof our model. This parameter scales the density peak nearthe interface until an objective macroscopic contact angle isobtained when rc → ∞ and N → ∞.

The model presented so far, although developed forgraphite, can be easily adapted to more complex structuressuch as silicon (cubic diamond). The anisotropic nature ofsilicon can be accounted for in the model through the ρsparameter and using Eq. (4) by properly shifting the interactionpotential contribution from different parallel atomic planes.Figure 1 illustrates the marked difference between the atomicstructures when the planes (100) and (111) are oriented on thex-y plane. For the case of Si(100), h0 = aSi/4, and for Si(111),the triple bilayer structure is described by h1 = aSi

√3/12 and

h2 = aSi√

3/4, where aSi is the lattice constant of silicon.For the case of silicon and other non-graphitic-like sub-

strates, it has been suggested to assume the solid as a semi-infinite block where the crystal structure normal to the wettedplane is characterized by an atomic volumetric density.21,28

It will be demonstrated that the accountability of the atomicstructure in the normal direction to the wetted plane is of greatimportance for a proper calculation of the contact angle.

III. MOLECULAR DYNAMICS MODELINGOF WETTABILITY AND POSTPROCESSING METHODS

A. Molecular dynamics model

The wettability of graphitic-carbon and silicon was inves-tigated using the cylindrical droplet method. A water box withperfectly arranged molecules was placed at a prudent distance

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

128.61.141.172 On: Fri, 24 Jul 2015 14:14:41

044703-4 Ramos-Alvarado, Kumar, and Peterson J. Chem. Phys. 143, 044703 (2015)

on top of a solid surface and the system was allowed to equil-ibrate until a water droplet was formed. Water was modeledby the extended simple point charge (SPC/E)29 model. Struc-turally speaking, the SPC/E model consists of a rigid structuremolecule with three point charges located at the oxygen(−0.8476e) and hydrogen (+0.4238e) positions interactingthrough a truncated Coulombic potential. Additionally, oxygenatoms interact through a LJ potential whereσOO = 3.166 Å andεOO = 0.65 kJ/mol. In order to keep the rigidity of the O–Hbonds (1 Å) and the H–O–H angle (109.47◦), the SHAKE30

algorithm was employed. The Coulombic interactions weretreated with the PPPM31 algorithm with an accuracy of 1×10−6.

The solid atoms were held fixed by setting the velocities tozero and restricting any force acting on them for the majorityof the simulations. For the cases of self-interacting solids, thebonded interactions between carbon and silicon atoms weretreated using Tersoff32 potentials, and in the case of carbon,interlayer interactions were modeled using a LJ potential withparameters σCC = 3.41 Å and εCC = 0.289 kJ/mol.33 Periodicboundary conditions were imposed on the three directions ofthe computational domain. The starting molecular setup of thesimulations is depicted in Fig. 2. The y-dimension of the solidsubstrates was fixed as 26.16 Å and 27.15 Å for graphite andsilicon, respectively. As for the x-dimension, the length variedbetween 200 Å and 300 Å depending on the size of the droplet;likewise, the z-dimension was kept sufficiently large to avoidany interaction between the water molecules of one box withthe periodic image of the solids of another.

The interaction between water molecules (oxygen atoms)and solid atoms was modeled using a truncated LJ pairwisepotential. A variety of energy and distance parameters weretaken from Ref. 10 for analysis. For the silicon simulations,Barisik and Beskok20 suggested σSi–O = 3.41 Å and εSi–O= 1.457 kJ/mol. The cutoff radius of the solid-water potentialwas varied as 10, 15, and 20 Å, and the cutoff for the Coulombicpotential was 15 Å. The LAMMPS34 code was used for the MDsimulations along with the VMD35 software for visualization.The time step for integration of the governing equations was1 fs, the neighbor lists were updated every time step, andthe center of mass of the water molecules was reset to itsinitial position every time step in order to avoid drifting dueto random perturbations. The simulation procedure was as fol-lows: (1) energy minimization of the structures in order to elim-inate any excess potential energy from the initial configuration;

FIG. 2. Initial molecular configuration of bi-layer graphene and a water boxwith 7920 molecules. (a) x-z plane view, (b) x-y plane view.

(2) equilibration at 298 K using a Nosé-Hoover thermostat,36

one for water and one for the solid atoms when required, witha time constant of 0.1 ps for 0.5 ns; (3) equilibration in themicrocanonical ensemble for 0.5 ns; (4) production run(3-4 ns) for collecting snapshots of the water molecules every0.5 ps.

B. Contact angle calculation

The computational box was discretized into bins of squarecross section in the x-z plane, see Fig. 2, and utilizing the entirelength of the computational box in the y-direction as the depthof the bins. The resolution of the bins was as large as 2 × 2 Åand as small as 0.125 × 0.125 Å. The position of each atom inthe water group was stored into the appropriate bin for everysnapshot and the mass density was calculated as the averageover time of the bins count per unit volume. Once the density ofthe droplet was obtained as a ρ(x, z), the sigmoidal function37

depicted in Eq. (9) was used to fit the density profile along thecenter of the droplet in the z-direction,

ρ (z) = 12(ρl + ρv) − 1

2(ρl − ρv) tanh

(z − ze

de/2

), (9)

where ρl is the bulk liquid density, ρv is the bulk vapor den-sity, ze is the position of the equimolar distance, and de isan approximation of the liquid-vapor interface thickness. ρvwas assumed zero and the data fit was carried out neglect-ing the highly distorted region near the solid surface.23 Thedroplet interface was defined by placing markers at the outerbins where ρ(x, z) = ρl/2, then the interface was refined usinga linear interpolation algorithm. The interface points wereadjusted by a circular fit based on the least squares method andthe contact angle was obtained from the slope at the intersec-tion with the solid surface. The contact angle calculation wasthe last step of the postprocessing stage and this operation wasperformed every ten snapshots allowing accumulation of thedata points over time. Hence, the evolution of the contact anglecalculation can be tracked as a function of the cumulative snap-shots used for the calculations. Additionally, the bulk densityand the root mean square error (RMSE) of the functional fits(sigmoidal and circular) were tracked over time.

Previous investigations have reported instantaneous calcu-lations of the contact angle as a function of time37 andsome others argued that the instantaneous calculation methodis more cost effective than averaging density profiles overtime.19,38 However, their own results illustrate ample oscil-lations of the instantaneous computations, and the standarddeviations of their steady state calculations are quite large. Byfollowing the multi-response postprocessing method (MRPM)described above, where several outcomes of the calculationsare tracked over time, a more comprehensive assessment ofthe postprocessing stage is accomplished obtaining precisecomputations. It will be demonstrated that a large number ofsnapshots are required for having a good quality calculation ofthe contact angle based on the statistics of the circular fit anddensity, instead of only looking at the contact angle evolutionover time.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

128.61.141.172 On: Fri, 24 Jul 2015 14:14:41

044703-5 Ramos-Alvarado, Kumar, and Peterson J. Chem. Phys. 143, 044703 (2015)

FIG. 3. MRPM analysis of a waterdroplet made of 4050 molecules withbinning resolution of 2×2 Å (blacklines), 1×1 Å (red lines), 0.5×0.5 Å(green lines), and 0.25×0.25 Å (bluelines): (a) contact angle, (b) bulk den-sity, (c) RMSE of the circular fit,(d) RMSE of the sigmoidal density fit.

IV. RESULTS AND DISCUSSION

A. Molecular simulations results

As a first step in the MD modeling of graphitic-carbonwettability and to show some of the features of the MRPM,one case study reported by Werder et al.10 was taken as areference. Three molecular setups made of 2376, 4050, and7920 molecules were simulated in a cylindrical droplet fashionwith fixed solid atoms (two layers of graphene) and using thesame LJ parameters (σCO = 3.19 Å and εCO = 0.3135 kJ/mol).Figure 3 illustrates the quality analysis of the contact anglecalculations as a function of the accumulated data sets overtime for a water droplet made of 4050 molecules using differentbinning resolutions. Each and every simulation was analyzedin the same fashion, but in the interest of brevity, only thecase depicted in Fig. 3 is set as an example of the MRPM.Figure 3(a) depicts the evolution of the contact angle overa postprocessing time of 3 ns. The contact angle calculationexhibits a rapid drop from 150◦ to 105◦ in a time frame ofapproximately 0.15 ns for every binning resolution analyzed.It is noteworthy that all the evolution curves follow similarpaths for different bin sizes. The inset of Fig. 3(a) depicts asubtle difference of approximately 1.25◦ from peak to peakalong the curves of maximum and minimum resolution, whichmay indicate a negligible effect of the bin size for this particularcase. However, there is a clear trend towards convergence to asingle curve as the binning resolution increases, leading to amore accurate calculation of the contact angle.

After gathering data sets for 0.5 ns, it appears that thecontact angle calculation converged. However, the MRPMprovides a set of quality indicators to be examined beforeperforming a final calculation. For example, the bulk value ofdensity obtained from the sigmoidal fit proposed in Eq. (9) wastracked and depicted in Fig. 3(b). Larger bins can enclose moreatoms and smoother density profiles are obtained towards aconverging value of 1 g/cm.3 On the other hand, the finest binresolution curve is the one that shows more noisiness comparedto the other density curves. For the particular case analyzedhere, it can be said that all density curves converged after1 ns. The range of variation of density is not significant inthis particular example; however, this variable is fundamentaland must be tracked. The most dramatic assessment of the

calculations quality comes from the evaluation of the RMSE ofthe circular fit used for representing the liquid-vapor interface.The RMSE was chosen over the least squares R2 parametersince the RMSE imposes much higher penalties to the qualityof a functional fit. It is demonstrated that although the contactangle curves begin showing steady state behaviors very rapidly,the quality of the circular fit operation is quite poor and signif-icantly affected by the bins size. By means of visual inspec-tion of several pictures of the circular representation of theinterface, a threshold value of RMSE ≤ 0.3 was adopted as anindicator of reliable calculations. Such a condition is eventu-ally achieved after gathering data sets for 1 ns. The last qualitydiscriminant investigated was the RMSE of the sigmoidal func-tion fit to the density profile along the centerline of the dropletdepicted in Fig. 3(d). Unlike the circular representation of thedroplet interface, the representation of the density profiles wasan easier task, due to the resemblance between Eq. (9) andthe MD-obtained density profiles. Similar to the circular fit,the RMSE of the density profile is affected by the binningresolution in the same fashion. Clearly, smaller bins are unableto capture good statistics as fast as larger bins.

Further investigation of the binning resolution is illus-trated in Fig. 4(a) for the same conditions described above.It can be clearly observed that a large bin size does not fullyrepresent the density profiles properly. Although smooth statis-tical results are obtained at the cost of fewer time steps, largebin sizes fail in capturing the fine details of the nanosystemsunder investigation. Similarly to Fig. 3(a), it is observed thatthe density profiles show a trend of convergence toward a singledescriptive curve. Adding to this discussion, Fig. 4(b) depictsno significant modification of the density profile curves bychanging the droplet size, neither in the relative position of thepeak densities nor in the magnitudes of such.

After evaluating all the information and taking intoconsideration the threshold values for the quality indicators(RMSE), the contact angle for the molecular setup whoseanalysis is reported in Fig. 3 was obtained by averagingbetween 1 ns and 3 ns. The resulting contact angle was106.01◦ ± 0.38◦ for an individual run and 106.25◦ ± 0.84◦ forthe average of three simulation runs under different initialconditions, all cases using the finest binning resolution. Thecontact angle calculations obtained by following the MRPM

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

128.61.141.172 On: Fri, 24 Jul 2015 14:14:41

044703-6 Ramos-Alvarado, Kumar, and Peterson J. Chem. Phys. 143, 044703 (2015)

FIG. 4. Density profiles along the center line of water droplets where z = 0 represents the solid substrate, lines are guides to the eye. (a) Binning resolutions of2×2 Å (circles), 1×1 Å (squares), 0.5×0.5 Å (diamonds), 0.25×0.25 Å (triangles). (b) Droplet size: 2376 molecules (left-facing triangles), 4050 molecules(right facing triangles), 7920 molecules (stars).

are remarkably accurate and repeatable results were obtainedafter analyzing different independent runs. Standard deviationsof up to 6.8◦ have been reported using the conventionalprocedure of averaging density profiles over time,13 andstandard deviations of 4◦ and higher have been observed bymeans of the instantaneous calculations of the contact angle.38

A feature of the MRPM is that it allows for a better evaluationof the progress of the contact angle calculations for eachparticular molecular setup, giving to the analysts the freedomto define threshold values or determining if longer samplingtimes are required. Extending this concept, the MRPM can beused to determine the appropriate time span for the productionruns for systems with a given size and number of liquidmolecules. Additionally, the starting point above which datasets can be used to average the contact angle calculation oncethe quality control variables reach a sought value can be easilydetermined. Different production runs can be found in theliterature for obtaining steady state contact angle calculationsfor wettability of water, for example, 0.2 ns for a systemwith 2526 molecules;39 0.5 ns for 322-5117 molecules,13 2744molecules,20 and 1000-8000 molecules;40 0.6 ns for 2000-8000molecules;10–12 1 ns17 and computationally wasteful produc-tion runs such as 100 ns for systems with 250-4000 mole-cules.21 We analyzed systems with 2000-8000 molecules,encompassing the range indicated above and found that datasampling runs as short as 0.6 ns are not able to produce reliableresults based on the uncertainty of the intermediate steps beforeactually calculating the contact angle and the wide amplitudeof the oscillations of the contact angle; thus, the MRPM can beused as a reference frame for similar investigations and thereis room for further optimization or addition of new qualityvariables. One last feature of the MRPM is that convergenceto a steady state calculation can be evaluated by selectingdifferent starting times for averaging data sets; this exampleis illustrated in supplementary Fig. S1 and an example of badstatistics in Fig. S2.44

Werder et al.10 calibrated the LJ oxygen-carbon potentialto retrieve θ∞ = 86◦ for graphite. After three trial tests usinga longer electrostatic potential cutoff radius, flexible graphite,and a different initial configuration of the water block, theyobserved slight variations of the resulting contact angle for asystem with 2000 molecules; thus, it was indicated that usingan oxygen-carbon potential cutoff radius rc = 10 Å was justi-fiable. In a follow-up investigation, Jaffe et al.11 found that rc

had a non-negligible effect for MD simulations of wettability.Convergent contact angle measurements were found for rc≥ 17.5 Å. As a consequence of these findings, new LJ parame-ters and the respective values of rc for obtaining θ∞ = 86◦werereported. Subsequent MD investigations of graphitic-carbonwettability have used the LJ parameters suggested in Ref. 10with rc = 10 Å,12,41 the LJ parameters suggested in Ref. 10 withrc = 20 Å,39,40 and some authors explored the influence of rcin their simulations.18,19,28 The effect of rc was investigated forthe particular condition previously analyzed by the MRPM andthe following pairs (rc, θ∞) were found (10 Å, 109.5◦ ± 0.49◦),(15 Å, 103.56◦ ± 0.38◦), and (20 Å, 103.17◦ ± 0.31◦). Rafieeet al.40 conducted the exact same simulation using a cylindricaldroplet with rc = 20 Å and they obtained an average contactangle of 109.96◦. Figure 5 illustrates these results and thecorresponding macroscopic contact angle extrapolations.

Two aspects can be highlighted about the MD modeling.First, the cylindrical method showed a droplet size independentbehavior in three out of the four cases reported in Fig. 5, asexpected from the modified Young’s equation. Such a behaviorwas observed by Weijs et al.15 and Peng et al.14 One caseshowed a small but perceptible size effect, a feature alsoreported by Wu and Aluru,18 but not as drastic as what wasobserved by Scocchi et al.13 Second, the linear relationshipthat exists between the parameter εCO and the contact anglefound by Werder et al.,10 Liu et al.,42 and Driskill et al.19

has been confirmed herein. Therefore, we took on the task of

FIG. 5. Calibration of the interaction potential for the prediction of graphitewettability and effects of cutoff radius. Symbology: Werder et al.10 rc= 10 Å(circles); same case as in Ref. 10, rc= 10 Å (squares); same case as in Ref. 10,rc= 15 Å (diamonds); stars and plus symbols for calibration studies withrc= 15 Å.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

128.61.141.172 On: Fri, 24 Jul 2015 14:14:41

044703-7 Ramos-Alvarado, Kumar, and Peterson J. Chem. Phys. 143, 044703 (2015)

FIG. 6. Snapshots of the molecular arrangement of water molecules on(a) Si(100) surface and (b) Si(111) surface.

calibrating our MD model to predict the most recently reportedvalue of θ∞ on a free-of-contaminants graphite surface byusing the linear relationship between εCO and the contact angle.We started by running a random hydrophillic simulation (εCO= 0.4974 kJ/mol while keeping σCO constant) to obtain asecond pair (εCO, θ∞)=(0.4974 kJ/mol,57.7◦ ± 0.24◦) in addi-tion to the one already found (0.3135 kJ/mol, 103.56◦ ± 0.38◦).Then, a linearly interpolated value of εCO = 0.4736 kJ/molwas determined for an objective θ∞ = 64.4◦.6 The results of theMD simulations under the new energy parameter are depictedin Fig. 5, where it can be observed that θ∞ = 64.7◦ ± 0.32◦.The calibration of the graphite contact angle to 64.4◦ wasfurther verified by running a simulation with only 2376 watermolecules (for simplicity and assuming size independence)interacting with three graphene layers. The resulting contactangle was 64.13◦ ± 0.14◦. Then, the same simulation wasrepeated allowing the graphene layers to interact and theresulting contact angle was 63.56◦ ± 0.26◦. In conclusion, itwas fully verified that the combination of parameters σCO= 3.19 Å, εCO = 0.4736 kJ/mol, and rc = 15 Å can be used toobtain θ∞ = 64.4◦ from MD simulations in graphitic surfaces.

The wettability analysis of Si(111) and Si(100) wassystematically conducted similar to that of graphite. Usingthe LJ parameters proposed by Barisik and Beskok,20 contactangles of 87.7◦ ± 0.36◦ and 103.6◦ ± 0.45◦ were obtained forSi(111) and Si(100), respectively (see Fig. 6 for snapshotsof the molecular setups in equilibrium). In their investigation,Barisik and Beskok20 used the hemispherical droplet method tocalibrate the LJ potential parameters to obtain θ∞ = 88◦. It hasbeen demonstrated here and in previous investigations that the

hemispherical and the cylindrical droplet methods converge inthe macroscopic limit. Interestingly, convergence was foundbut not for the same silicon plane. From a theoretical point ofview, it could be expected for the Si(111) surface to be morewettable than the Si(100) surface due to the marked differencein their atomic surface density (as suggested in Section II) ifthe water molecules and silicon atoms interacted through thesame pairwise potential. Similar observations about the contactangle dependence on the substrate atomic density have beenreported.43 Despite being an issue, these results opened a doorof opportunity for testing our analytical model with differentatomic structures.

Barisik and Beskok20 observed some degree of entrain-ment of water molecules into the interstitial of the Si(100) sur-face through snapshots of the molecular setup in equilibrium;they also reported density profiles but the resolution of theircontours was very poor. The density contours in Fig. 7(a) prop-erly captured the entrainment of water molecules in every in-terstitium of the Si(100) plane and Fig. 7(b) depicts a snapshotof the water molecules penetrating the large spaces between Siatoms in the plane (100). Such a phenomenon was not observedfor the Si(111) surface nor the graphitic surfaces whose atomicsurface densities are 7.83 nm−2 and 37.24 nm−2, respectively,whereas the atomic density of the Si(100) plane is 6.78 nm−2.Further implications of these differences will be discussed inSec. IV B.

B. Theoretical model results

Shih et al.21 used a MF model to get fundamental insightinto the wettability of single-layer graphene. Their model onlyconsidered long-range attractive forces, neglected interfacialentropy, and assumed the water particles Boltzmann distrib-uted. The denominator of the attractive term of the LJ potentialwas the only fitting parameter of their model, which was foundby adjusting their model to predict a θ∞ = 86◦ on graphite. Wefollowed a similar approach by neglecting the entropy contri-bution in Eq. (6) and tuning the parameter η. A comparisonof the revised model of Shih et al.21 and ours is depicted inFig. 8 for the same target contact angle. We found θ∞ = 84◦

after N = 100 using their model and parameters, whereas ourmodel reaches θ∞ = 86◦ after N = 6. Both models predict thecontact angle and ∆UWS/A magnitudes very closely, but ourcalculations are closer to MD simulations in terms of the N

FIG. 7. (a) Density contours of a water droplet made of 2460 water molecules and (b) close-up of the entrainment phenomenon observed on the Si(100) surfacedue to its small atomic surface density.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

128.61.141.172 On: Fri, 24 Jul 2015 14:14:41

044703-8 Ramos-Alvarado, Kumar, and Peterson J. Chem. Phys. 143, 044703 (2015)

FIG. 8. Comparison between our model (filled symbols) and Shih et al.21

(open symbols).

required to reach the macroscopic limit. Our model indicatesthat at N = 3, 97% of the macroscopic value of ∆UWS/A isreached; similarly, our MD calculations of the contact angleare found to reach the macroscopic limit after N = 3. Shihet al.21 found that the contact angle of graphene was 96◦ buttheir MD simulations reported 104◦. Our model, on the otherhand, predicts a contact angle of graphene of 99.8◦, a closervalue to their MD result.

The model of Shih et al.21 is significantly affected by theutilization of a purely attractive interactive potential such thatno potential well exists and the potential rapidly diverges toinfinity. Additionally, the assumption of having water mole-cules Boltzmann distributed significantly overpredicts the wa-ter molecules count per unit area (NW/A) near the wall causingthe model to be highly biased towards hydrophilic outcomes.Thus, in order to avoid the density distribution and potential todiverge to infinity, they had to define a parameter δGL = 3.28 Åas the position of the density peak away from the surface sothat the integration of Eq. (5) could be finite. However, ourMD simulations and others have found δGL ≈ σCO for grapheneand this distance slightly decreases as more carbon layers areadded.23 The calibration process performed here was for amacroscopic contact angle obtained via MD simulations withLJ parameters σCO = 3.19 Å and εCO = 0.392 kJ/mol.10 Theresult of such a calibration on our model goes beyond forcinga desired output for a given condition but to predict other MD-

derived results for different values of εCO as it will be shownshortly.

Table I summarizes a comparison of NW/A obtainedbetween zmin = 0.22 nm and zmax = 0.88 nm using differentsources for the representation of ρL(z) such as MD simu-lations (NMD

W /A), the SKA (NSKAW /A), the Boltzmann distri-

bution (NBW/A), and our Boltzmann-calibrated model (BCM)

(NBCMW /A); these calculations were performed on a bi-layer

graphene system like in Ref. 24 and with rc = 2 nm. Taherianet al.24 suggested to use the SKA of ρL(z) for the calculationof T∆SWS/A for obtaining NW/A values closer to MD-derivedcalculations. However, the application of the SKA is verysensitive to the value of zmin and there is no clear rule abouthow to define it. The utilization of a Boltzmann distributiondoes not require a careful definition of zmin, but a signifi-cant overprediction of NW/A is observed when compared tothe MD data. Interestingly, the calibration performed in ourmodel resulted in a good match to MD data as shown inTable I.

Next, we calculated the contact angle for the range ofεCO values listed in Table I. The calculations were performedusing the BCM (neglecting entropy) and the BCM includ-ing the entropy term in the calculation of Wa, referred to asthe BEM. MD and analytical calculations of the contact an-gle were taken from Ref. 24 for comparison purposes. TheMD simulations were conducted using the cylindrical dropletmethod and rc = 20 Å; therefore, the MD contact angles canbe considered close to θ∞ for each case. The analytical re-sults come from a MF model using the SKA for representingthe liquid density. Taherian et al.24 acknowledged that theirinterfacial entropy model was only suitable under hydrophobicconditions when compared to MD simulations. However, thefull analytical application of this model in the BEM resultsin a poor match between analytical and MD contact angleseven at εCO ≈ 0.2 kJ/mol, see Fig. 9. One can argue that thelack of accuracy of the BEM could be improved if the modelwas calibrated to match a given contact angle, just like theBCM. Such a calibration process was conducted (BECM) andresulted in forcing the model to overcompensate the parameterNW/A as shown in the last column of Table I. As a consequence,

TABLE I. Comparison between NW/A calculated using different ρL(z) functions between the limits zmin andzmax as a function of εCO.

εCO (kJ/mol) NMDW /A a (nm−2) NSKA

W /A b (nm−2) NBW/A c (nm−2) NBCM

W /A d (nm−2) NBECMW /A (nm−2)

0.0940 18.6 22.4 25.1 22.0 22.70.1254 19.5 22.0 26.9 22.1 23.10.1567 20.3 22.0 28.9 22.2 23.70.1881 20.8 22.0 31.4 22.3 24.30.2508 21.6 21.6 37.3 22.5 25.60.3135 22.2 21.6 45.3 22.8 27.00.3762 22.7 21.6 56.2 23.1 28.80.4389 23.0 21.6 70.9 23.5 30.70.5016 23.3 21.2 91.3 23.9 32.80.5643 23.6 21.2 119.5 24.3 35.20.6270 23.8 21.2 158.8 24.7 37.9

aValues obtained from MD simulations data.24

bValues obtained using the SKA.24

cCalculation using the Boltzmann distribution.dCalculation calibrating our model to a macroscopic contact angle of 86◦.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

128.61.141.172 On: Fri, 24 Jul 2015 14:14:41

044703-9 Ramos-Alvarado, Kumar, and Peterson J. Chem. Phys. 143, 044703 (2015)

FIG. 9. Calculation of the contact angle for different values of εCO: MDdata (circles),24 BCM (squares), BEM (triangles), and the SKA-based model(diamonds).24

the values of ∆UWS/A and T∆SWS/A are overcompensated tooand significantly deviated from MD data, see Fig. S3 of thesupplementary material.44

Analytical models that neglect the entropy contribution ofthe work of adhesion and use the SKA have been employedfor predicting the contact angle dependence on temperatureand their limitations have been pointed out.28 Taherian et al.24

indicated that SKA-based models coincide with MD calcula-tions for contact angles close to 86◦ using σCO = 3.19 Å andεCO = 0.392 kJ/mol, because at that point, the underpredictionof∆UWS/A matches Wa due to neglecting T∆SWS/A, see Fig. 9.The BCM features a better match with MD-derived contactangles as depicted in Fig. 9. The calibration process conductedfor a single contact angle, as indicated above, adjusts the modelto closely represent the trend of ∆UWS/A but not so close thatentropic contributions become significant, see Fig. S3 of thesupplementary material.44 Thus, the BCM poses an alternativefor MF-based models that require calibration for the predictionof contact angle and other variables derived from it.

The BCM showed to be versatile when applied to sili-con. First, the water monomer potential calculation had to beadjusted to the normal direction of the wetted plane followingthe strategy discussed in Section II. The model was calibratedto a θ∞ = 88◦ on Si(111). The corresponding analytical calcu-lation for Si(100) was 106.95◦ very close to the MD calculationof 103.6◦. Now if the model is calibrated to match θ∞ = 103.6◦

on Si(100), the resulting contact angle of the Si(111) surfaceis 81.2◦. From the BCM point of view, the triple bilayer ofthe silicon structure normal the plane (111) poses a significantdifference with respect to graphite. Additionally, both silicon

FIG. 10. Comparison between MD and the BCM calculations of θ∞: Werderet al.10 (circles), Scocchi et al.29 (diamonds), Taherian et al.23 (crosses),Taherian et al.24 (stars), and our MD results for graphite (squares) and silicon(triangles). Not all data points were available with error bars.

planes studied here have different atomic densities. It is thenimportant to make a remark about the capabilities of our modelto capture all these fine details in the calculation of the macro-scopic contact angle for different atomic structures.

Figure 10 depicts a comparison between the BCM withMD contact angle calculations gathered throughout thisinvestigation. Simulation results obtained using LJ potentialparameters proposed by other authors to predict different θ∞values of graphitic surfaces as well as our current calibrationto 64.4◦ were used as inputs to our BCM, in addition to thecorresponding data points for silicon surfaces. It must be indi-cated that the analytical values of θ∞were obtained for N → ∞and rc → ∞. Overall, a good match between MD simulationsand our analytical model was found. Even though the BCMrequires a different calibration for graphite and silicon, its capa-bilities to match MD simulations should be highlighted sincethis is something not observed in previous investigations21,28

proposing similar models. The main difference arises in a betteraccountability of the density profiles of liquid particles byscaling the Boltzmann distribution function and the lack of azmin or any other minimum distance fitting parameter requiredto evaluate the energy interaction integral, see Eq. (5).

The comparisons between the BCM, BEM, and SKA-based models were conducted using N = 2 for similarity pur-poses. However, while the MD simulations of graphite wetta-bility reach their size-convergence at N = 3, analytical modelsdo not show such a behavior, see Fig. 8. In order to obtainmacroscopic properties, N must be large enough; for example,the calculation of Eb of graphite is affected by N and

FIG. 11. (a) Binding energy and (b) contact angle of graphite as a function of the number of graphene layers and rc= 10 Å (circles), rc= 15 Å (squares),rc= 25 Å (triangles), rc→ ∞ (solid line).

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

128.61.141.172 On: Fri, 24 Jul 2015 14:14:41

044703-10 Ramos-Alvarado, Kumar, and Peterson J. Chem. Phys. 143, 044703 (2015)

FIG. 12. Contact angle as a function of the number of graphene layers for σCO= 3.19 Å and εCO= 0.392 kJ/mol (filled symbols), and σCO= 3.0 Å andεCO= 0.45 kJ/mol (open symbols); (a) similar binding energies produce different contact angles; (b) the difference in the contact angles is more accuratelyrelated to ∆UWS/A.

the magnitude of rc. Werder et al.10 indicated that θ∞ = 86◦

for graphite could be retrieved from MD simulations if Eb= −6.3 kJ/mol. We calculated Eb as a function of N for severalvalues of rc, see Fig. 11(a), and found Eb = −6.29 kJ/mol inthe limit when N → ∞ and rc → ∞ for a distance betweencarbon atoms of 1.44 Å. More recently, Driskill et al.19 foundEb = −6.44 kJ/mol which is a value that can be obtained ifthe distance between carbon atoms is 1.42 Å. As indicatedabove, the observed behavior of the contact angle in MDsimulations is different from the analytical one in terms ofN . The continuum assumption and liquid density distribu-tion approximations feature the most significant limitationswhen comparing the MD and analytical models. Figure 11(b)depicts a series of calculations to assess the effect of rc onthe contact angle of graphite using the BCM under the sameconditions studied in the validation case in Section III. Besidesthe differences between MD and the BCM, similarities arisein the sense that increasing rc leads to a reduction of thecontact angle, also analytically reported by Sendner et al.28 Thefollowing results were found for rc = 10 Å, θMD

∞ = 109.5◦ andθBCM∞ = 110.6◦; rc = 15 Å, θMD

∞ = 103.5◦ and θBCM∞ = 106.1◦;

rc = 20 Å, θMD∞ = 103.3◦ and θBCM

∞ = 104.4◦. In the limit ofrc → ∞, the analytical contact angle was 103.1◦, a value closeto the MD calculation for rc ≥ 15 Å. It must be highlighted thatMD simulations with rc = 15 Å showed size dependence andthe contact angle was obtained from extrapolation using themodified Young’s equation; however, the average of the contactangle using different droplet sizes was ∼105◦, very close to theanalytical value. Because of these fine but existing differences,we recommend to compare MD and analytical models in themacroscopic limit. The simulations from Ref. 24 were con-ducted using N = 2 and rc = 20 Å; thus, the deviations fromthe macroscopic state were not significant.

The significance of Eb as a fundamental variable for aprediction of the contact angle was questioned by Wu andAluru.18 They determined Eb using density functional theory(DFT) calculations and accounted for different orientationsof the water monomer. The DFT-based obtained potentialswere fitted to be used in MD simulations and these werefound to radically underpredict the contact angle of wateron graphite. Additionally, they observed that potentials thatpredicted similar binding energies when used in MD simu-lations resulted in contact angle differences of 8◦. We inves-tigated this issue from a theoretical point of view using theBCM. Figure 12(a) illustrates how a different combination

of interaction potentials can result in similar Eb but differentθ∞. Figure 12(b) presents a better explanation to the observeddifference in the contact angles. If both LJ parameters arechanged, the water density distribution is altered, namely, therelative position between the first density peak and the solidsurface as well as the magnitude of NW/A. More importantly,a significant change in the water monomer potential wN(z)can be expected to eventually modifying ∆UWS/A. Pairwiseinteraction potentials are calibrated via MD simulations for agiven surface at fixed temperature and rc; therefore, Eb lacksof generality to be correlated with the contact angle whenthe calibrated interaction potential changes, except when σCOremains constant. In addition, Eb clearly correlates linearlywith εCO as it can be observed from Eqs. (4) and (5). Un-questionably, a more comprehensive manner to account forthe changes produced in the molecular setup after changingthe interaction potential is ∆UWS/A (from a MF theory pointof view) or the work of adhesion as implied from the veryfoundations of thermodynamics.23

V. SUMMARY AND CONCLUSIONS

Graphitic carbon and silicon surfaces wettability has beeninvestigated via MD simulations and theory. Guidelines havebeen presented for an appropriate calculation of the contactangle from data obtained via MD simulations. Methods fortracking the evolution of the contact angle as well as the qualityof the intermediate steps performed during the evaluation ofthe contact angle have been suggested (MRPM). The MRPMfeatures an improvement in the accuracy and reliability inthe calculations of the contact angle as compared to previousinvestigations. Likewise, the inappropriateness of the limitedpostprocessing times reported in previous investigations hasbeen highlighted. A calibration of the water-carbon pairwiseinteraction potential was conducted for obtaining a macro-scopic contact angle of 64.4◦ on graphite. Appropriate dimen-sions and cutoff radius for the potentials were suggested forfuture MD simulations.

An analytical model based on the MF theory was devel-oped assuming that water and solid substrates interact througha LJ potential including attractive and repulsive forces. Theconventional MF approach and the interfacial entropy contri-bution were considered for the calculation of the work ofadhesion. The full model accounting for interfacial entropychanges was found useful only in a very limited range of highly

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

128.61.141.172 On: Fri, 24 Jul 2015 14:14:41

044703-11 Ramos-Alvarado, Kumar, and Peterson J. Chem. Phys. 143, 044703 (2015)

hydrophobic surfaces. A calibration to the conventional MFmodel neglecting entropy was conducted in the density distri-bution function to match a given macroscopic contact angleobtained from MD simulations. The calibrated model showeda remarkable capability for predicting graphite contact anglesunder different wettability conditions as well as the wettabilityof two different silicon planes. Just like models based on theSKA, our calibrated model compensates the lack of account-ability for entropy changes with a lower prediction of energyinteractions. The calibrated model proposed here also served todemonstrate that Eb is not as significant as previously thoughtwhen calculating MD wettability. In summary, the calibrationmethod proposed here is an alternative to SKA-based methodsfor analytical calculations of wettability.

ACKNOWLEDGMENTS

While performing this work, Bladimir Ramos-Alvaradowas supported by the Mexican Council on Science and Tech-nology (CONACyT) under Scholarship No. 312756.

1F. M. Fowkes and W. D. Harkins, J. Am. Chem. Soc. 62, 3377 (1940).2I. Morcos, J. Colloid Interface Sci. 34(3), 469 (1970).3M. E. Schrader, J. Phys. Chem. 84(21), 2774 (1980).4S. R. Wang, Y. Zhang, N. Abidi, and L. Cabrales, Langmuir 25(18), 11078(2009).

5Y. J. Shin, Y. Y. Wang, H. Huang, G. Kalon, A. T. S. Wee, Z. X. Shen, C. S.Bhatia, and H. Yang, Langmuir 26(6), 3798 (2010).

6Z. T. Li, Y. J. Wang, A. Kozbial, G. Shenoy, F. Zhou, R. McGinley, P. Ireland,B. Morganstein, A. Kunkel, S. P. Surwade, L. Li, and H. T. Liu, Nat. Mater.12(10), 925 (2013).

7C. J. Shih, M. S. Strano, and D. Blankschtein, Nat. Mater. 12(10), 866 (2013).8M. J. P. Nijmeijer, C. Bruin, A. F. Bakker, and J. M. J. Vanleeuwen, Phys.Rev. A 42(10), 6052 (1990).

9F. Leroy, D. J. V. A. dos Santos, and F. Muller-Plathe, Macromol. RapidCommun. 30(9-10), 864 (2009).

10T. Werder, J. H. Walther, R. L. Jaffe, T. Halicioglu, and P. Koumoutsakos, J.Phys. Chem. B 107(6), 1345 (2003).

11R. L. Jaffe, P. Gonnet, T. Werder, J. H. Walther, and P. Koumoutsakos, Mol.Simul. 30(4), 205 (2004).

12J. H. Walther, T. Werder, R. L. Jaffe, P. Gonnet, M. Bergdorf, U. Zimmerli,and P. Koumoutsakos, Phys. Chem. Chem. Phys. 6(8), 1988 (2004).

13G. Scocchi, D. Sergi, C. D’Angelo, and A. Ortona, Phys. Rev. E 84(6),061602 (2011).

14H. Peng, G. R. Birkett, and A. V. Nguyen, Mol. Simul. 40(12), 934 (2014).

15J. H. Weijs, A. Marchand, B. Andreotti, D. Lohse, and J. H. Snoeijer, Phys.Fluids 23(2), 022001 (2011).

16M. Brinkmann, J. Kierfeld, and R. Lipowsky, J. Phys.: Condens. Matter17(15), 2349 (2005).

17D. Sergi, G. Scocchi, and A. Ortona, Fluid Phase Equilib. 332, 173(2012).

18Y. B. Wu and N. R. Aluru, J. Phys. Chem. B 117(29), 8802 (2013).19J. Driskill, D. Vanzo, D. Bratko, and A. Luzar, J. Chem. Phys. 141(18),

18C517 (2014).20M. Barisik and A. Beskok, Mol. Simul. 39(9), 700 (2013).21C. J. Shih, Q. H. Wang, S. C. Lin, K. C. Park, Z. Jin, M. S. Strano, and D.

Blankschtein, Phys. Rev. Lett. 109(17), 176101 (2012).22E. S. Machlin, Langmuir 28(49), 16729 (2012).23F. Taherian, V. Marcon, N. F. A. van der Vegt, and F. Leroy, Langmuir 29(5),

1457 (2013).24F. Taherian, F. Leroy, and N. F. A. van der Vegt, Langmuir 29(31), 9807

(2013).25K. C. Daoulas, V. A. Harmandaris, and V. G. Mavrantzas, Macromolecules

38(13), 5780 (2005).26W. A. Steele, Surf. Sci. 36(1), 317 (1973).27J. N. Israelachvili, Intermolecular and Surface Forces, 3rd ed. (Academic

Press, 2011), 704 pp.28C. Sendner, D. Horinek, L. Bocquet, and R. R. Netz, Langmuir 25(18), 10768

(2009).29H. J. C. Berendsen, J. R. Grigera, and T. P. Straatsma, J. Phys. Chem. 91(24),

6269 (1987).30J.-P. Ryckaert, G. Ciccotti, and H. J. C. Berendsen, J. Comput. Phys. 23(3),

327 (1977).31R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles

(CRC Press, 1988), p. 540.32J. Tersoff, Phys. Rev. B 39(8), 5566 (1989).33M. Shen, P. K. Schelling, and P. Keblinski, Phys. Rev. B 88(4), 045444

(2013).34S. Plimpton, J. Comput. Phys. 117(1), 1 (1995).35W. Humphrey, A. Dalke, and K. Schulten, J. Mol. Graphics 14(1), 33 (1996).36S. Nose, Mol. Phys. 52(2), 255 (1984); W. G. Hoover, Phys. Rev. A 31(3),

1695 (1985).37M. J. de Ruijter, T. D. Blake, and J. De Coninck, Langmuir 15(22), 7836

(1999).38D. Vanzo, D. Bratko, and A. Luzar, J. Chem. Phys. 137(3), 034707 (2012).39R. Raj, S. C. Maroo, and E. N. Wang, Nano Lett. 13(4), 1509 (2013).40J. Rafiee, X. Mi, H. Gullapalli, A. V. Thomas, F. Yavari, Y. F. Shi, P. M.

Ajayan, and N. A. Koratkar, Nat. Mater. 11(3), 217 (2012).41X. Y. Li, L. Li, Y. Wang, H. Li, and X. F. Bian, J. Phys. Chem. C 117(27),

14106 (2013).42J. Liu, C. L. Wang, P. Guo, G. S. Shi, and H. P. Fang, J. Chem. Phys. 139(23),

234703 (2013).43T. A. Ho, D. V. Papavassiliou, L. L. Lee, and A. Striolo, Proc. Natl. Acad.

Sci. U. S. A. 108(39), 16170 (2011).44See supplementary material at http://dx.doi.org/10.1063/1.4927083 for

more information about the MRPM and the analytical model.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

128.61.141.172 On: Fri, 24 Jul 2015 14:14:41