International Journal of Heat and Mass Transfer 59 (2013) 144–154

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

A molecular dynamics study on transport properties and structure at theliquid–vapor interfaces of alkanes

Hari Krishna Chilukoti ⇑, Gota Kikugawa, Taku OharaInstitute of Fluid Science, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan

a r t i c l e i n f o a b s t r a c t

Article history:Received 20 July 2012Received in revised form 9 October 2012Accepted 4 December 2012Available online 7 January 2013

Keywords:Molecular dynamics simulationLiquid–vapor interfaceStructureSurface tensionSelf-diffusion coefficientAlkane

0017-9310/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.12

⇑ Corresponding author. Address: Molecular Heat TrFluid Science, Tohoku University, 2-1-1 Katahira, AobaTel./fax: +81 22 217 5872.

E-mail address: chilukoti@microheat.ifs.tohoku.ac.

In this paper, self-diffusion and structure in the vicinity of liquid–vapor interfaces of several species ofn-alkanes are examined using molecular dynamics simulations. The united atom NERD force field wasutilized to model the alkane molecules, and the obtained properties were validated by the reportedexperimental results. Surface tension was calculated based on the averaged pressures, and tail correc-tions were implemented based on the molecular virial formulae. Present surface tension results forbutane, hexane and decane are in good agreement with experimental and other simulated values. Molec-ular orientation and chain conformation in the interface region were examined by observing the orderingparameter and radius of gyration for decane and tetracosane. In the interface region, molecules arepreferentially oriented parallel to the interface and shape of the molecules is slightly flattened in thez-direction for both decane and tetracosane. With an increase in temperature, molecules are preferen-tially less parallel to the interface. The Einstein relation modified for confined regions was used to obtainthe self-diffusion coefficient according to migration of molecules parallel to the interface. Self-diffusioncoefficient increases in the interface region for both decane and tetracosane. This increase in self-diffusion coefficient has a correlation with decrease in density. The dependency of self-diffusion coeffi-cient on density is nonlinear in the liquid and vapor side of the interface region and is linear in betweenthe above two regions. Influence of chain length of the alkane molecules has been studied by comparingthe physical and structural quantities of decane, tetracosane and hexatriacontane at the same reducedtemperature. At the same reduced temperature, self-diffusion coefficient decreases with an increase inchain length in the interface region as well as in the bulk liquid region. All three alkanes show similartendency of ordering and chain conformation at the interface at the same reduced temperature.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction Transport properties and structure of the materials at the interface

Liquids of hydrocarbon chain molecules have many industrialapplications such as lubrication and coating, and are major build-ing blocks for long polymers. Due to their essential importancefor many technological and biological processes, liquid interfaceshave attracted the attention of many scientists. Studies of thestructure and transport properties in the vicinity of interfaces playan important role in understanding of adhesion, wetting and lubri-cation. Coating films with a thickness of the order of nanometershave been realized and are coming into industrial use [1–3].Understanding of equilibrium and non-equilibrium properties isalso important for such nanoscale materials, since their coatingflow behavior and quality of the residual films are dominated bysurface effects. When the thickness of the polymeric films is innano meters, macroscopic physics does not work for these films.

ll rights reserved..015

ansfer Laboratory, Institute of-ku, Sendai 980-8577, Japan.

jp (H.K. Chilukoti).

region are different from that of bulk liquid region. This is relatedto the difference in the interaction between the molecules in thebulk liquid and in the interface region. The molecular-scale struc-ture of the system will influence the properties of the nanoscalesystem. So their knowledge is important to realize optimal thermo-physical and mechanical properties. To reproduce the molecular-scale structure and transport phenomena and to analyze theircharacteristics, molecular dynamics (MD) simulation is the bestsuitable method and a number of studies have been reported forpolymers with reliable inter- and intramolecular potential models.

Surface tension is one of the basic properties related to free en-ergy of liquid–vapor interfaces and has an important role in engi-neering applications like oil and water repellency, wetting, andadhesion. It tells about the chemical nature of the component andinfluences its affinity towards other components like wettability.Surface tension of some n-alkane liquids at several temperatureswere calculated in the present study paying attention to the influ-ence of long range intermolecular potential energies, and comparedwith other existing data to validate the present simulations.

Nomenclature

A total surface areabeq equilibrium bond lengthCVk particular slabD self-diffusion coefficientkh intramolecular force parameter for bendingkr intramolecular force parameter for bond stretchingL length of simulation boxN number of moleculesP pressureH ordering parameterS survival probabilityr distance between two atomsRg radius of gyrationU potential energy

Greek symbolsc surface tensiond angle between the connecting line joining two carbon

units and interface normale energy parameter for the Lennard–Jones potential

model

h bond anglek interface thickness measured by fitting the number den-

sity profile to the hyperbolic tangent functionq densityq(z) density distribution along z-direction# polar angler size parameter for the Lennard–Jones potential models time intervalu dihedral angle

Subscriptc cut-off radiusCH2 methylene parameterCH3 methyl parameterij between units i and jcr critical temperaturep surface tension value during simulationtail tail correction contributionx, y, z x, y and z coordinates0 equilibrium state

H.K. Chilukoti et al. / International Journal of Heat and Mass Transfer 59 (2013) 144–154 145

Diffusion is one of the fundamental properties of liquids, whichis related to mass transport. Molecular transport in liquids in thevicinity of liquid–vapor interfaces is important in various processessuch as mixing, adsorption to the interface, evaporation and con-densation. Polymer molecules change their shapes and orienta-tions through their migration process, which is reflected on themass transport properties. Since these changes of molecular struc-ture have influences on other transport properties such as viscosityand heat conduction, the behavior of mass transport will give somesuggestions also to other transport phenomena in the liquid–vaporinterface regions. Where the experimental measurements are diffi-cult, some new insights will be brought by MD simulations. Anearly study of self-diffusion of small alkanes in the liquid statewas reported by a few researchers [4,5]. The main drawback oftheir work was that they considered only small amount of timeinterval in the calculation of self-diffusion coefficient. In the lastdecade diffusion of alkane in the liquid state was studied by manyresearchers [6–13] using MD simulations with isotropic and aniso-tropic united atom models. Few researchers tested [4–9] the repro-ducibility of the existing models for alkane molecules for transportproperties and some other researchers modified the existing forcefields for better reproducibility of transport properties in liquidphase [10–13]. Self-diffusion in binary liquid n-alkane mixtureswas studied by Harmandaris et al. [8]. Concerning the self-diffu-sion in the liquid–vapor interface region, MD studies for waterhas been reported [14–17]. To the best of the authors’ knowledge,the self-diffusion of n-alkanes in the liquid–vapor interface regionhas not been reported in the literature. In this paper, the self-diffu-sion in the direction parallel to the interface was studied for liquiddecane (C10H22), tetracosane (C24H50) and hexatriacontane (C36H74)in the vicinity of the interface. Effect of chain length on propertieshas also been studied by comparing them at the same reducedtemperature.

2. Computational details

2.1. Molecular models and potential functions

Mainly three models have been utilized to describe an alkanemolecule for molecular dynamic simulations, which include the

all atom (AA) model [18], the united atom (UA) model [19] andthe anisotropic united atom (AUA) model [13]. In the UA model,each of CH2 and CH3 groups is represented by a pseudoatom, whichis connected to other pseudoatoms to represent linear chain mole-cules. The deformation of this linear chain is defined by stretchingbetween two pseudoatoms, bending defined by three neighboringpseudoatoms, and torsion defined by four neighboring pseudoa-toms. The AUA is a sort of UA, although in the AUA model, the forcecenter is moved away from the carbon for better accounting ofinteractions. The AA models reproduce the molecular structureincluding hydrogen atoms and flexibility of molecules related tothem, although the computational time is significantly higher thanthe UA models. Mainly for computational reasons, most of theresearchers used UA in studies of alkane properties. Over the years,several variants of force fields were developed to study propertiesof alkanes [7–13,18–24]. Among those, the NERD force field wasused by a few researchers [8,23,25–27] to simulate properties ofpure n-alkanes and their binary mixtures. In the present work,we used a united atom model to describe alkane molecules. Fromnow on we refer to the united pseudoatoms as atoms. The NERDforce field [25] was used to define intra- and intermolecular inter-actions. In this force field, total potential energy of the system iswritten as the sum of nonbonded interactions, bond bending, bondstretching and torsion potential. In the NERD force field, interac-tion between two atoms which are located more than three atomsapart on the same molecule and atoms located on the differentmolecules are given by the Lennard–Jones (LJ) 12-6 potential

ULJðrijÞ ¼ 4eijrij

rij

� �12

� rij

rij

� �6" #

; ð1Þ

where rij is the distance between atoms i and j. The energy param-eters for the methyl and methylene groups areeCH3 ¼ 1:4358� 10�21 J and eCH2 ¼ 6:3233� 10�22 J, respectively.The size parameter of CH3 and CH2 units are rCH3 ¼ 3:91 Å andrCH2 ¼ 3:93 Å, respectively. Interactions between two different sitesare described by the standard Lorentz–Berthelot combining rules

eij ¼ffiffiffiffiffiffiffiffiffieiiejj

pand rij ¼

rii þ rjj

2; ð2Þ

The LJ interaction was truncated with a cut-off radius of 13.8 Å.

(a)

(b)

Position-z (A)

Den

sity

(kg/

m3 )

-100 -50 0 50 100

200

400

600

800 T=350KT=400KT=450K

Decane

o

800T=400KT=450KTetracosane

146 H.K. Chilukoti et al. / International Journal of Heat and Mass Transfer 59 (2013) 144–154

The bond bending, bond stretching and torsion are the deforma-tions defined by other interaction potential functions includedwithin the chains. The bending takes place between any threeneighbor atoms within a molecule and is modeled by the harmonicpotential

UðhÞ ¼ 12

khðh� h0Þ2; ð3Þ

where h is the bond angle. The constant kh is 8.6291 � 10�19 J/rad2

and equilibrium angle h0 is 114.0�. The bond stretching potential isgiven as follows:

UðrijÞ ¼kr

2ðrij � beqÞ2; ð4Þ

where kr is 1.3323 � 10�18 J/Å2 and beq is 1.54 Å. The torsion occursbetween any four neighboring atoms in a molecule. The torsion po-tential that works between two atoms which are two atoms apart isdefined by the following function

UðuÞ ¼ V0 þ V1ð1þ cos uÞ þ V2ð1� cos 2uÞ þ V3ð1þ cos 3uÞð5Þ

where u is the dihedral angle, and V0 = 0 J, V1 = 4.9018 � 10�21 J,V2 = �9.4146 � 10�22 J and V3 = 1.0925 � 10�20 J.

2.2. Computational systems

To study the structure and transport properties at the interface,we have performed the molecular dynamic simulations of liquid–vapor equilibrated systems. The time integration of equations ofmotion is implemented by the reversible reference system propa-gator algorithm (r-RESPA) method [28] with multiple time scales(MTS). An integration time step of 1 fs for the intermolecular mo-tion and 0.2 fs for the intramolecular motion were used in all sim-ulations. Periodic boundary conditions and minimum imageconvention were applied in three spatial directions. The simulationsystem was maintained at constant temperatures using Nose-Hoo-ver thermostat. The simulation system is shown in Fig. 1. A liquidfilm is enclosed in a box of lengths Lx = Ly = 50 Å and Lz = 240 Å fordecane, Lx = Ly = 70 Å and Lz = 320 Å for tetracosane and Lx =Ly = 100 Å and Lz = 400 Å for hexatriacontane. In order to avoidthe interaction of molecules present at the two surfaces throughthe vapor phase, generally, Lz was taken as 3–4 times larger thanLx and Ly. The sizes of the systems investigated in this work rangedfrom 500 molecules for both decane and tetracosane to 1000 mol-ecules for hexatriacontane. In the present work, initial configura-tion was created by placing the molecules randomly at thecenter of the rectangular parallelepiped simulation box to form aliquid film and thus enclosed by vacuum. After that, the systemtemperature was gradually raised to the required temperatureand then simulations were performed for NVT ensemble, i.e., with

Fig. 1. MD simulation system of liquid–vapor interface.

constant number of particles, volume and temperature for 6 � 106

intermolecular time steps. After reaching the equilibrium state, thesystem had two planar interfaces normal to the z axis and thick-ness of the liquid film was changed to attain the corresponding li-quid densities at that temperature. Equilibrium averages werecollected for about 2 ns.

3. Results and discussion

3.1. Density profiles

All quantities were calculated after the system has reachedequilibrium and interface has been formed. Before studying struc-tural quantities and transport properties, the liquid densities ob-tained in our simulations have been validated by comparingthem with literature. Density profiles for decane at T = 350 K,400 K and 450 K, and tetracosane at T = 400 K, 450 K, and 500 Kare shown in Fig. 2. The local density was obtained by dividingthe rectangular parallelepiped simulation box into slabs with athickness of 1 Å in the z direction. Because liquid films in the sim-ulation box move due to thermal agitation at higher temperatures,it has been advised to plot the density profile based on the centerof mass of the system [24]. In the present paper, the density pro-files are plotted with respect to the instantaneous center of massof the system. The density profile exhibits a flat region of a well-equilibrated bulk liquid phase at the center of the simulation boxcoexisting with vapor phases at both sides. For decane, the presentliquid density (624 kg/m3 at 400 K) is slightly lower than the liquid

Position-z (A)

Den

sity

(kg/

m3 )

-100 0 1000

200

400

600

T=500K

o

Fig. 2. Density distributions for (a) decane at T = 350 K, 400 K and 450 K, and (b)tetracosane at T = 400 K, 450 K and 500 K.

(a) (b)

(c)

Position-z(A)

Scal

ed n

umbe

r den

sity

-20 -15 -10 -5 0 5 10

5E+26

1E+27

1.5E+27

2E+27

2.5E+27

3E+27

CH3CH2Allatom

Decane at 350K

oPosition-z(A)

o

Position-z(A)o

Scal

ed n

umbe

r den

sity

-20 -10 0 10

5E+26

1E+27

1.5E+27

2E+27

2.5E+27

CH3CH2All atom

Decane at 400K

Scal

ed n

umbe

r den

sity

-20 -10 0 10

5E+26

1E+27

1.5E+27

2E+27

2.5E+27

CH3CH2All atom

Decane at 450K

Fig. 3. Scaled number density distributions for decane at (a) 350 K, (b) 400 K and (c) 450 K.

H.K. Chilukoti et al. / International Journal of Heat and Mass Transfer 59 (2013) 144–154 147

density (652 kg/m3) reported by Nath et al. [25] for the NERD dec-ane. The cut-off radius for the LJ interactions may influence on thedensity. The density profiles are symmetric about the center ofmass of the system. As expected, Fig. 2 shows that with an increasein temperature, liquid density decreases and vapor density in-creases. Distributions of scaled number densities of CH3, CH2 andall atoms for decane and tetracosane at three temperatures areshown in Figs. 3 and 4, respectively. The number densities arescaled to see to it that the values of the three number densitiesare identical in bulk liquids. For example, a decane molecule hastwo CH3 groups and eight CH2 groups. Based on this, the numberdensities of CH3 and CH2 are scaled by the factors of 1/2 and 1/8,respectively. In the figure, a significant difference is observed be-tween CH3, the end group of alkanes, and CH2. The number densityof CH2 is lower in the liquid side of the interface region and it ishigher in the vapor side as compared with the distribution ofCH2 and all atoms. This observation suggests that molecules exhi-bit a certain orientation. The orientation of molecules in the vicin-ity of the interface region will be discussed further in the nextsection based on the analysis of the orientation order parameter.

The 10–90 interfacial thickness is the distance over which thetime-averaged density changes from 90% of its liquid density to10% of its liquid density [20]. The 10–90 interfacial thicknessesfor different alkanes at several temperatures are shown in Table 1.Here, interfacial thickness k is measured by fitting the time-aver-aged number density distribution along the z-axis to the hyper-bolic tangent function [29,30], and is related to the usual 10–90interfacial thickness as 4.394 k. It is observed that as temperaturegoes higher the interfacial thickness is larger for an alkane.

Molecular dynamics predictions of the critical points of decane,tetracosane and hexatriacontane (C36H74) are 617 K, 805 K and862 K by Nath et al., respectively [25]. Fig. 5 shows the density dis-tributions along the z-axis for decane, tetracosane and hexatria-contane at a reduced temperature 0.7Tcr. At the same reducedtemperature (0.7Tcr), all the tested alkanes have nearly equal liquiddensity. Vapor density decreases with increase in chain length.

3.2. Molecular-scale structures at the interface

The orientation order parameter is used here to quantify the ef-fect of interface on chain ordering. It can be experimentally deter-mined using infrared-visible sum frequency spectroscopytechniques [18]. A convenient orientation order parameter is de-fined in the MD system based on the second order Legendre poly-nomial as follows [31]

HðzÞ ¼ 12h3 cos2 d� 1i; ð6Þ

where d is the angle between a vector connecting two carbon atoms,which are two units apart in a molecule, and interface normal, thatis, z-axis. The average h i is taken over all vectors which are pre-sented in the specified slab, which will be described below, for anentire production run. This ordering parameter ranges from �0.5to 1. The magnitude of ordering parameter zero indicates com-pletely random arrangement of molecules. A positive value indi-cates that molecules are preferentially perpendicular to theinterface, while a negative value indicates that molecules are pref-erentially parallel to the interface. The values of 1 and �0.5 indicate

(a) (b)

(c)

Position-z (A)

Scal

ed n

umbe

r den

sity

-30 -20 -10 0 10

2E+26

4E+26

6E+26

8E+26

1E+27

1.2E+27

1.4E+27

CH3CH2Allatom

Tetracosane at 400

o Position-z (A)

Scal

ed n

umbe

r den

sity

-30 -20 -10 0 10

2E+26

4E+26

6E+26

8E+26

1E+27

1.2E+27

CH3CH2Allatom

Tetracosane at 450

o

Position-z (A)

Scal

ed n

umbe

r den

sity

-30 -20 -10 0 10

2E+26

4E+26

6E+26

8E+26

1E+27

1.2E+27

CH3CH2Allatom

Tetracosane at 500

o

Fig. 4. Scaled number density distributions for tetracosane at (a) 400 K, (b) 450 K and (c) 500 K.

Table 110–90 Thickness of liquid–vapor interfaces for several n-alkanes at severaltemperatures.

Molecule Temperature(K)

10–90 Interfacial thickness(Å)

Decane (C10H22) 350 9.4365400 14.1996432(0.7Tcr) 14.2167450 17.9648

Tetracosane (C24H50) 400 9.2063450 11.2227500 13.7365563.5(0.7Tcr) 17.0887

Hexatriacontane (C36H74) 603.4(0.7Tcr) 18.9385

Position-z (A)

Den

sity

(kg/

m3 )

-200 -100 0 100 200

200

400

600

800C10H22 at 432KC24H50 at 563.5KC36H74 at 603.4K

o

Fig. 5. Density distributions as a function of the position along z-axis for decane,tetracosane and hexatriacontane at 0.7Tcr.

148 H.K. Chilukoti et al. / International Journal of Heat and Mass Transfer 59 (2013) 144–154

that molecules are completely normal and parallel to the interface,respectively. To calculate spatial distribution of the orientation or-der parameter, the simulation box was divided into slabs with awidth of 1 Å in the z direction. A vector connecting between twoatoms was allocated to the slab where the midpoint between thetwo atoms was located.

Fig. 6 shows the orientation order parameter for decane andtetracosane at three temperatures. Density distributions are alsoshown in dashed lines in the figures. These data and transportquantities are presented by taking the Gibbs dividing surface asthe origin. The position of the Gibbs dividing surface was deter-mined by fitting the time-averaged number density distributionsalong the z axis to the hyperbolic tangent function [29]. The plots

for the two interfaces in the present system are practically identi-cal, so only the data for the right interface is shown in the figure.

Fig. 6 illustrates that the molecules are oriented randomly in thebulk liquid region. In respect to the orientation parameter, thevicinity of interface region consists of three layers. In the innermost layer close to the bulk liquid, the orientation order parameteris slightly positive, which suggests that molecules are orientedpreferentially perpendicular to the interface. In the middle layer

(a) (b)

Position-z (A)

Orie

ntat

ion

orde

r par

amet

er

Den

sity

(kg/

m3 )

-30 -20 -10 0 10

-0.06

-0.04

-0.02

0

0.02

0.04

0

100

200

300

400

500

600

700

T=350KT=400KT=450K

decane

Orientation order parameter

Density

o

(c)

Position-z (A)

Orie

ntat

ion

orde

r par

amet

er

Den

sity

(kg/

m3 )

-30 -20 -10 0 10

-0.1

-0.05

0

0.05

0.1

0

200

400

600

T=400KT=450KT=500K

Tetracosane

Density

Orientation order parameter

o

Orie

ntat

iono

rder

para

met

er

-30 -20 -10 0 10

-0.06

-0.04

-0.02

0

0.02

0.04

0

100

200

300

400

500

600

700

o

Decaneat350K

Orientationorderparameter

Density

Position-z(A)

Den

sity

(kg/

m3 )

Fig. 6. The orientation order parameter as a function of distance for (a) decane at T = 350 K, 400 K and 450 K, (b) decane at T = 350 K with the range of a standard deviation, (c)tetracosane at T = 400 K, 450 K and 500 K. The density profiles are shown in dashed lines.

Position-z (A)

Orie

ntat

ion

orde

r par

amet

er

Den

sity

(kg/

m3 )

-40 -20 0

-0.02

-0.01

0

0.01

0.02

0

100

200

300

400

500

C10H22 at432KC24H50 at563.5KC36H74 at603.4K

Orientation order parameter

Density

o

Fig. 7. The orientation order parameter as a function of distance for decane,tetracosane and hexatriacontane at 0.7Tcr. The density profiles are shown in dashedlines.

H.K. Chilukoti et al. / International Journal of Heat and Mass Transfer 59 (2013) 144–154 149

next to the above layer, with a density in the middle of the liquidand vapor, the order parameter is negative, which suggests thatmolecules are oriented preferentially parallel to the interface. Inthe outer most layer close to the vapor side, the orientation orderparameter increases rapidly to a positive value again. It is consid-ered that the parameter decreases and approaches zero towardsthe bulk vapor phase, although reliable data for vapor phase wasnot obtained in the present simulations due to the small numberof samples. To indicate the uncertainty of the ordering parameters,standard deviation of the data has been checked for decane at350 K at several locations in the simulated region. A set of theinstantaneous data during 2 ns was divided into 10 subsets andthen, an averaged value of each subset was obtained; the averageand standard deviation of these 10 values were calculated andthe range of average values plus and minus the standard deviationis indicated in Fig. 6b. It is observed from the figure that the uncer-tainty of the data near the vapor phase is relatively larger thanthose in the liquid and interface regions. This is resulted becauseof the less sampling number. The orientation order parameter inthe vapor phase was examined by another MD simulation ofhomogeneous bulk vapor phase of decane at 350 K and 0.878 kg/m3, and it was verified that the orientation order parameter is zeroin the vapor phase. Both decane and tetracosane show similar kindof alignment in the interface region. This tendency of orientationsuggests that the rotation of molecules in the immediate vicinityof the interface is restricted [31], and similar kinds of trends werealso reported by MD simulations of alkanes for different force fields

[18,31]. With an increase in temperature, these preferential orien-tations are less obvious for both decane and tetracosane. At a par-ticular temperature, longer alkanes exhibit more preferentialorientation. Fig. 7 shows the orientation order parameter for threealkanes at the same reduced temperature. All three alkanes shownearly the same tendency.

Position-z (A)

Com

pone

nt o

f Rg2

(m2 )

Den

sity

(kg/

m3 )

-30 -20 -10 0 10

1E-19

1.5E-19

2E-19

0

200

400

600

800

T=400KT=450KT=500K

Rg2 inx-direction

Rg2 inz-direction

Tetracosane

o

Fig. 9. Components of radius of gyration as a function of distance for tetracosane atT = 400 K, 450 K and 500 K. The density profiles are shown in dash dotted lines.

150 H.K. Chilukoti et al. / International Journal of Heat and Mass Transfer 59 (2013) 144–154

Radius of gyration is another manifestation of the chain confor-mation. In the present study, mean-square radius of gyration R2

g

has been calculated using the following formula [32]

R2g ¼

1N2

XN�1

i¼1

XN

j¼iþ1

ð~Ri �~RjÞ2* +

; ð7Þ

where N is the number of atoms in the molecule. The vector R!

i � R!

j

is one connecting all the pairs of sites belonging to the same mole-cule. To calculate the distribution of the radius of gyration in thesystem, the simulation box was divided into slabs with a thicknessof 3 Å in the z direction and the average of Eq. (7) was taken in eachslab. Typical distributions of the total radius of gyration across theliquid–vapor interface are shown in Fig. 8, in which the total radiusof gyration is plotted as functions of the z-axis for tetracosane atthree different temperatures. The total radius of gyration is constantin the bulk liquid region and slightly decreases in the interface re-gion, which suggests that the molecular shape depart slightly froma linear one in the interface region. The value of the total radius ofgyration for tetracosane in the vapor phase at T = 500 K andq = 0.578 kg/m3 is 4.11 � 10�19 m2, which was determined by aMD simulation of homogeneous bulk vapor phase. Although this va-lue cannot be acquired by the liquid–vapor interface simulation dueto the small number of samples in the vapor phase, it is consideredthat the value in the liquid phase, 4.24 � 10�19 m2 at T = 500 K, de-creases across the interface region and approaches 4.11 � 10�19 m2

in the vapor phase.The vector R

!i � R!

j in Eq. (7) was decomposed to the x-, y- andz-components to examine the difference between normal and par-allel components to the interface. The mean-square radii of gyra-tion in the directions normal and parallel to the interface asfunctions of the z-axis are presented in Fig. 9. Density distributionis also shown in the figure. It can be seen from the figure that thecomponent of mean square radius of gyration in the normal direc-tion decreases significantly in the interface region while those forthe parallel directions increase. From Fig. 6 it is known that mole-cules are preferentially parallel to the interface in the middle of theinterface region. Based on these facts from Figs. 6 and 9, it can beconcluded that molecules flatten in the z direction than in otherdirections. The variation in the interface region is smaller at highertemperature. Since the molecular shape is only slightly changed inthe interface region, as was seen in Fig. 8, it is concluded that thisremarkable variation in each of the components of radius of gyra-tion is a direct results of the change in molecular orientation in the

Position-z (A)

Tota

l Rg2

(m2 )

Den

sity

(kg/

m3 )

-40 -20 00

2E-19

4E-19

6E-19

8E-19

0

200

400

600

800

T=400KT=450KT=500K

Tetracosane

Density

Radius of gyration

o

Fig. 8. Total radius of gyration as a function of distance for tetracosane at T = 400 K,450 K and 500 K. The density profiles are shown in dashed lines.

interface region. It was observed by the homogeneous bulk vaporphase simulation mentioned above, all the components of themean-square radii of gyration are equally 4.11 � 10�19 m2 in thebulk vapor phase. It is expected that the values showing remark-able changes in the interface region in Fig. 9 converge to this valuetoward the vapor phase.

3.3. Surface tension

Surface tension of alkanes at liquid–vapor interfaces was calcu-lated by many researchers using molecular dynamics simulations[18–24]. To calculate surface tension accurately from moleculardynamics simulations, one needs to properly consider the numberof molecules, lateral size of the system, and perpendicular distanceto the interphase. The other important issue is proper treatment oflong range correction in the inhomogeneous region. In the past fewyears, many researchers used different techniques [19,20,23,24,33,34] to consider the full interaction in the surface tension calcu-lation. Each method has its own merits and demerits over othermethods. In the present paper, we used the method suggested byMecke et al. [34] for the tail correction in surface tension calcula-tions of alkanes, which is described here briefly.

Kirkwood-Buff formula was applied to calculate the surface ten-sion ðcPÞ during the simulation in a rectangular parallelepiped sim-ulation box, which is expressed by the following formulae [23,20]

cP ¼Lz

2hPzzi �

hPxxi þ hPyyi2

� �; ð8Þ

where hPzzi and ðhPxxi þ hPyyiÞ=2 are the averaged normal and tan-gential components of pressure. Lz is the length of the simulationbox in the z direction. Presence of two interfaces is taken into ac-count by the factor ½. The surface tension, cP , calculated form Eq.(8) does not include the nonbonded potential interactions beyondthe cut-off radius, rc. To take into account the effect of the truncatedLJ potential, the long-range correction should be added to the cP . Inthis work, tail corrections were implemented using the method out-lined by Mecke et al. [34]. In this method, the tail contribution fromthe individual particles, i, was calculated at each time step and thenform the average

ctail ¼1

2A

Xi; r>rc

ctaili

* +; ð9Þ

where A = 2LxLy denotes the total surface area. For the united atomof CH3, the tail contributions from other CH2 and CH3 atoms, ctail

iðCH3Þ

Table 2Comparison of surface tension values for several alkanes with experimental values and reported simulation results with the NERD model.

Molecule Temperature (K) Present work (mN/m) ðcP þ ctail ¼ cÞ Experimental value [23] (mN/m) NERD [23] (mN/m)

Butane (C4H10) 295 8.8 + 4.4 = 13.2 11.8 13.3Hexane (C6H14) 350 8.8 + 4.0 = 12.5 12.5 14.7Decane (C10H22) 350 13.8 + 5.5 = 19.3 18.6 20.8

400 9.3 + 4.2 = 13.5 14.4 16.5450 6.9 + 2.8 = 9.7 10.55 11.9432 (0.7Tcr) 7.6 + 3.6 = 11.2 N/A N/A

Tetracosane (C24H50) 400 16.5 + 6.9 = 23.4 N/A N/A450 13.9 + 5.8 = 19.7 N/A N/A500 10.5 + 4.8 = 15.3 N/A N/A563.5 (0.7Tcr) 6.0 + 3.5 = 9.5 N/A N/A

Hexatriacontane (C36H74) 603.4 (0.7Tcr) 6.6 + 3.5 = 10.1 N/A N/A

H.K. Chilukoti et al. / International Journal of Heat and Mass Transfer 59 (2013) 144–154 151

and ctailiðCH2Þ

, respectively, were calculated individually by the follow-ing equations and added together

ctailiðCH3Þ ¼ �24p

Z 1

r¼rc

dreCH3 2rCH3

r

� �12� rCH3

r

� �6� �

r2Z p

0d#ð1

� 3 cos2 #Þ sin#qCH3ðzi þ r cos#Þ

� 24pZ 1

r¼rc

dreCH3�CH2 2rCH3�CH2

r

� �12� rCH3�CH2

r

� �6� �

� r2Z p

0d#ð1� 3 cos2 #Þ sin#qCH2

ðzi þ r cos#Þ:

ð10Þ

ctailiðCH2Þ ¼ �24p

Z 1

r¼rc

dreCH3�CH2 2rCH3�CH2

r

� �12� rCH3�CH2

r

� �6� �

� r2Z p

0d#ð1� 3 cos2 #Þ sin#qCH3

ðzi þ r cos#Þ

� 24pZ 1

r¼rc

dreCH2 � 2rCH2

r

� �12� rCH2

r

� �6� �

r2

�Z p

0d#ð1� 3 cos2 #Þ sin#qCH2

ðzi þ r cos#Þ: ð11Þ

In Table 2, surface tension ðc ¼ cP þ ctailÞ of decane obtained in thepresent simulation with the corresponding tail corrections at sev-eral temperatures are compared with surface tension values ob-tained by Mendoza et al. [23] for a system with the NERD forcefield and with experimental values available in the literature. Inthe work of Mendoza et al. [23], the full LJ interaction betweenthe atoms were treated with the Ewald summation, which is com-putationally time demanding over the present method. As we cansee from the Table 2, present simulations over-predict the surfacetension of decane at lower temperature, and the error becomessmaller at higher temperatures. The simulation results of Mendozaet al. [23] are always higher than the experimental values. Surfacetension values are also compared for hexane and butane in Table 2.Overall, present results are in good agreement with the results ofMendoza et al. [23] and experimental values. Neglecting the long-range correction beyond the cutoff radius may result surface ten-sions that are far too low by as much as 40% [24]. It is concludedthat the appropriate tail corrections in the present study leaded tothe good results. Another new finding is that at the same reducedtemperature of 0.7Tcr, all the tested alkanes have surface tensionvalues close to 10 mN/m.

3.4. Self-diffusion coefficient in the interface region

The Einstein relation [8–10,35,36] or Green–Kubo relation[9,36] are generally utilized to determine self-diffusion coefficientfrom molecular dynamics simulations. To apply the Einstein rela-tion, the mean square displacement (MSD) of center of mass of

the molecules is acquired as a function of time over a long periodof time. By applying the Green–Kubo relation, self-diffusion coeffi-cient is calculated from the velocity auto-correlation. These meth-ods are originally suitable for homogeneous bulk systems. Todetermine the local value of MSD in a region and obtain the spatialdistribution of the self-diffusion coefficient, special treatment isneeded concerning the selection of molecules to be counted.Recently, Liu et al. [16] proposed a method to determine the self-diffusion coefficient based on the Einstein relation in such a case,which is applicable here for the interface region. In the presentpaper we applied this method to illustrate the variation ofself-diffusion coefficient in the liquid–vapor interface region ofdecane and tetracosane. The outline of the method is presentedhere following the discussion by Wick and Dang [17]. The basic cellof the molecular dynamics simulation was divided to theslabs with a certain thickness in the direction of z. The survivalprobability of a molecule to stay within a slab was determined asfollows:

SðsÞ ¼ Nð0; sÞNð0Þ ; ð12Þ

where Nð0; sÞ is the number of molecules which stay in the slab(CVk) continuously during the entire period of time s, and Nð0Þ isthe number of molecules present in the slab at the beginning. TheMSD in the direction of x observed in the specified slab after thetime period of s, which is averaged over the molecules that werepresent in the specified slab at the beginning, was calculated

hDxðsÞ2iCVk¼ 1

Nð0ÞX

i2CVk

ðxiðsÞ � xið0ÞÞ2; ð13Þ

For the summation of Eq. (13), only the molecules which stay in theslab CVk continuously during the entire period of time s arecounted. Then, the self-diffusion coefficient in respect to molecularmotions in the direction parallel to the interface is determinedusing the modified formula of the Einstein relation as follows:

DxxðCVkÞ ¼hDxðsÞ2iCVk

2sSðsÞ : ð14Þ

The self-diffusion coefficient parallel to the interface in a confinedregion can be calculated with this formula. To capture the variationof the self-diffusion coefficient across the interface, one needs fineslabs. On the other hand, the numbers of molecules present in theslab depends upon the thickness of the slab, and for a long periodof times, small number of molecules will survive in a slab duringthe tracking process of MSD. Thus, there should be a compromisefor the good results between the period of time to track the migra-tion of molecules and thickness of the slab.

Prior to the investigation of the variation of the self-diffusioncoefficient across the interface, we have validated our result for

Table 3Comparison of self-diffusion coefficient values for several alkanes with literature.

Molecule Temperature(K)

Density(kg/m3)

Literature Diffusioncoefficient(m2/s)

Hexane (C6H14) 275 642.8 Present 4.1 � 10�09

N/A AUA3 [12] 3.8 � 10�09

N/A AUA4 [12] 3.7 � 10�09

676.51 Expt. [38] 3.0 � 10�09

Decane (C10H22) 298 727.2 Present 1.9 � 10�09

724.7 A [11] 1.6 � 10�09

724.7 B [11] 1.4�10�09

N/A Expt. [11] 1.4 � 10�09

497.5 756.7 Present 3.4 � 10�09

754.1 UA [10] 2.7�10�09

754.1 AUA [10] 3.4 � 10�09

N/A Expt. [10] 2.6�10�09

Dodecane (C12H26) 372 692.7 Present 3.1 � 10�09

690.7 MSD [9] 2.8 � 10�09

690.7 VAC [9] 3.0 � 10�09

403.5 634.4 Present 5.0 � 10�09

N/A NERD [8] 4.5 � 10�09

152 H.K. Chilukoti et al. / International Journal of Heat and Mass Transfer 59 (2013) 144–154

bulk liquids. Reproducibility of the NERD potential for the self-diffusion coefficient was tested by comparing the present valueswith other predictions and experimental values at different con-ditions, which are shown in Table 3. The self-diffusion coefficientin the bulk liquid phase is calculated from homogeneous bulkliquid simulations. In most cases where the self-diffusion coeffi-

(a)

Position-z (A)

Den

sity

(kg/

m3 )

-50 -40 -30 -20 -10 0 100

100

200

300

400

500

600

12345601 9 8 7

o

(c)

Ti

MSD

par

alle

l to

the

inte

rfac

e (m

2 )

0 2E-0

5E-20

1E-19

1.5E-19

2E-19

2.5E-19

3E-19

in bin i

Decane at 40

Fig. 10. For decane at 400 K: (a) division of slabs on the density profile, (b) survival probafunction of time for several slabs.

cient in the bulk liquid phase is determined by MD simulations,bulk liquid simulations with the Einstein relation are utilized. Inthe present work, the modified Einstein relation was utilized tocalculate self-diffusion coefficient in liquid alkanes. The simula-tion box was divided into slabs and self-diffusion coefficientwas calculated in each slab, which gave nearly the same valuesas the values calculated from the value measured in a wholesimulation box using the Einstein relation. This was a checkfor the distribution of self-diffusion coefficient in the interfacesimulations, and the same method was applied for the interfacesimulation to obtain the distribution of self-diffusion coefficientin the liquid–vapor interface region. In addition to the liquidcases summarized in Table 3, a simulation of saturatedvapor for decane at 450 K and 4.378 kg/m3 was also performed.Self-diffusion coefficient value calculated from the Einstein rela-tion in the vapor phase is 2.76 � 10�6 m2/s, which is in goodagreement with theoretical predictions (2.64 � 10�6 m2/s) usingthe equation suggested by Zhu et al. [37]. These comparisonssuggest in total that the present results are in fair agreementwith the reported simulation results using different force fieldsand with experimental values. Although the NERD force field ap-plied in the present study has been optimized to reproduce equi-librium densities of alkanes and not for the self-diffusioncoefficient, the present results exhibit a fair reproducibility com-parable to other force fields such as UA and AUA models ofPadilla and Toxvaerd [10] and a UA model of Mondello and Grest[11] which have been optimized to reproduce self-diffusioncoefficient.

(b)

Time (s)

Surv

ival

pro

babi

lity

0 5E-12

0.2

0.4

0.6

0.8

1

slab # 8-10slab # 7slab # 6slab # 5slab # 4slab # 3slab # 2slab # 1

Decane at 400K

me (s)12 4E-12

ulk liquid regionnterface

0KSlab # 1

2

345678~10

bilities as a function of time for several slabs and (c) mean square displacement as a

Position-z (A)

Self-

diffu

sion

coe

ffici

ent (

m2 /s

)

Den

sity

(kg/

m3 )

-80 -60 -40 -20 0 200

5E-09

1E-08

1.5E-08

2E-08

2.5E-08

3E-08

0

100

200

300

400

500

600

C10H22 atT=432KC24H50 atT=563.5KC36H74 atT=603.4K

o

Fig. 12. Self-diffusion coefficient Dxx as a function of distance for decane, tetrac-osane and hexatriacontane at 0.7Tcr. The density profiles are shown in dashed lines.

s)

2E-08

H.K. Chilukoti et al. / International Journal of Heat and Mass Transfer 59 (2013) 144–154 153

To capture the variation of self-diffusion coefficient across theinterface, slabs with a thickness of 3 Å was applied. If smaller slabsare taken, the molecules quit the slabs within shorter time which isnot sufficient to get the slope in the MSD versus time curve. Typicalplots of MSD of center of mass of decane molecules versus time at400 K are shown in Fig. 10. Several MSDs obtained in the slabs bothin bulk liquid regions and interface regions are plotted in the fig-ure. The location of the slabs and the survival probabilities for sev-eral slabs are also shown. Remarkable varieties are observedamong the lines for the interface region, while no significant differ-ence is observed among the lines for the slabs close to the bulk li-quid. Variation of self-diffusion coefficient concerning molecularmigration in the direction parallel to the interface as a functionof the relative location to the dividing surface for decane andtetracosane are shown in Fig. 11 for three different temperatures.Each rectangular mark is plotted at the center of the slab. For allthe temperatures, the diffusion coefficient is constant in bulk liquidregion and increases toward the vapor phase in the interface re-gion. The self-diffusion coefficient increases with an increase intemperature for a particular alkane. Fig. 12 shows the self-diffusioncoefficient corresponding to the molecular migration in the direc-tion parallel to the interface for three alkanes at the same reducedtemperature. Bulk liquid alkanes with larger molecular chain

(a)

(b)

Position-z (A)

Self-

diffu

sion

coe

ffici

ent (

m2 /s

)

Den

sity

(kg/

m3 )

-60 -40 -20 0 200

1E-08

2E-08

3E-08

4E-08

5E-08

0

200

400

600

800

Dxx atT=350KDxx atT=400KDxx atT=450K

Decane

o

Position-z (A)

Self-

diffu

sion

coe

ffici

ent (

m2 /s

)

Den

sity

(kg/

m3 )

-60 -40 -20 0 200

5E-09

1E-08

1.5E-08

2E-08

0

200

400

600

800

Dxx atT=400KDxx atT=450KDxx atT=500K

Tetracosane

o

Fig. 11. Self-diffusion coefficient Dxx as a function of distance for (a) decane atT = 350 K, 400 K and 450 K and (b) tetracosane at T = 400 K, 450 K and 500 K. Thedensity profiles are shown in dashed lines and the self-diffusion coefficient profilesare shown in solid lines with rectangular squares.

Density (kg/m3)

Self-

diffu

sion

coe

ffici

ent (

m2 /

0 200 400 600 8000

5E-09

1E-08

1.5E-08 T=400KT=450KT=500K

Tetracosane

Fig. 13. Self-diffusion coefficient Dxx versus local density for tetracosane atT = 400 K, 450 K and 500 K.

length exhibit smaller self-diffusion coefficient, and this tendencyholds in the interface region.

The increase in the self-diffusion coefficient in the interface re-gion occurs at the same time with the decrease in density. The cor-relation between the self-diffusion coefficient and density isplotted in Fig. 13 for tetracosane at three temperatures. It is ob-served in the figure that the correlation is not linear in the regionsclose to the bulk liquid side and vapor side, while an almost linearcorrelation is observed in between. The linear correlation can beexplained as the effect of mixing of the bulk liquid characteristicsand those of bulk vapor, since the liquid and the vapor appearsalternately in this range due to the fluctuation of intrinsic surfaceat the scale of interface thickness. In contrast to this, the nonlinearcorrelation in the regions close to the liquid and vapor sides maybe an appearance of characteristics peculiar to the interface. Itshould also be noted that the triply layered structure found inthe distribution of the orientation order parameter in the interfaceregion is observed again.

4. Conclusion

Structure and mass diffusion properties in the vicinity of liquid–vapor interface of several linear chain n-alkanes were studied by

154 H.K. Chilukoti et al. / International Journal of Heat and Mass Transfer 59 (2013) 144–154

the molecular dynamics simulations using the NERD force field.Surface tension was carefully acquired with the tail correctionsand in general, present surface tension values are in good agree-ment with experimental and literature values. The orientation or-der parameter across the interface region exhibits that theorientation of alkane molecules are random in bulk liquids, prefer-entially normal to the interface in the closely liquid side of theinterface region, preferentially parallel to the interface in the mid-dle of the interface region, and again normal in the vapor side ofthe interface region. Thus, it was demonstrated that the liquid–va-por interface region of alkanes takes on triply layered structure.This variation of the orientation is more noticeable at lower tem-peratures. Chain conformation is studied by examining the radiusof gyration. At a given temperature molecules are flattened in thez-direction and shape of the molecules are slightly changed inthe interface region. The self-diffusion coefficient correspondingto the migration of molecules in the direction parallel to the inter-face, especially its variation across the interface region, is reportedhere for the first time for the NERD alkanes. The self-diffusion coef-ficient increases rapidly toward the vapor side, responding to thedecrease in density. The correlation between the self-diffusioncoefficient and the density in the interface region suggests thatthe interface region is divided into three regions. In the regionsclose to the liquid side and the vapor side, the correlation betweendensity and self-diffusion coefficient is nonlinear, while it is almostlinear in between.

If the results of different alkanes are compared at the same re-duced temperature, the orientation order parameter and the self-diffusion coefficient exhibited a similar variation in the interfaceregion over all the tested species of alkanes. The magnitude ofthe self-diffusion coefficient is different according to the chainlength of molecules, i.e., shorter molecules exhibit larger self-diffu-sion coefficient, in contrast to the fact that magnitude of surfacetension is almost equal when compared at the same reducedtemperature.

Acknowledgements

The work reported in this paper was supported by the Grant-in-Aid for Scientific Research and the 21st Century COE Program‘‘International COE of Flow Dynamics’’ by the Japan Society forthe Promotion of Science (JSPS). Numerical simulations were per-formed on the SGI Altix UV1000 at the Advanced Fluid InformationResearch Center, Institute of Fluid Science, Tohoku University.

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