molecular dynamics simulation of surface energy and zddp effects on friction in nano-scale...

Tribology International 43 (2010) 1811–1822

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Tribology International

0301-67

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/triboint

Molecular dynamics simulation of surface energy and ZDDP effects onfriction in nano-scale lubricated contacts

Hassan Berro �, Nicolas Fillot, Philippe Vergne

Universite de Lyon, CNRS, INSA-Lyon, LaMCoS UMR5259, F-69621, France

a r t i c l e i n f o

Article history:

Received 5 October 2009

Received in revised form

29 January 2010

Accepted 23 February 2010Available online 1 March 2010

Keywords:

Molecular dynamics

Super thin film lubrication

Additives

Surface energy

9X/$ - see front matter & 2010 Elsevier Ltd. A

016/j.triboint.2010.02.011

esponding author.

ail address: [email protected] (H. Ber

a b s t r a c t

Molecular dynamics simulations are used to study the tribological performance of a lubricant mixture

containing hexadecane base oil and 5% zinc dithiophosphate (ZDDP) under molecular confinement

conditions. The influence of ZDDP additive and the surface–lubricant interaction on the mechanical and

thermal interfacial response are studied in detail. Results show that mechanical and thermal slips are

reduced by increasing the surface energy. Simulations also demonstrate the migration of ZDDP

molecules and their adsorption onto the solid surface resulting in a remarkable suppression of

mechanical slip compared to pure hexadecane. Consequently, the effective shear rate is higher and so is

the friction.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The modern design of mechanical systems shows a growinginterest for the use of operating fluids as lubricants. Due to theirlow viscosities, under moderate operating conditions of load andshear, these lubricants can be confined down to the molecularscale. Generally speaking, molecularly confined films are sus-pected to be present in contact sub-regions of systems operatingunder ‘‘mixed lubrication’’. Friction coefficient in this lubricationregime increases as the operating conditions become more andmore severe and the lubricant film threatens to break down indifferent zones of the contact. Applications such as modernrolling-element bearings, micro-motors, and optical read/writedevices operate either partially or completely under mixedlubrication and thus the understanding of molecularly confinedflow becomes of increased importance.

In applications where nano scale film thicknesses are predictedand unless the contacting surfaces are specifically treated to bemolecularly smooth, the lubricant film is expected to break downin contact zones where the asperities interact [1]. In the regionswhere most lubricant molecules are squeezed out, anti-wearadditive protection layers resist and remain adsorbed to thesurfaces. The additive layers reduce the risk of direct metal–metalcontact hence achieving a very positive tribological role againstsurface adhesion, wear, and eventually global friction. Inother zones of the same lubricated contact, it can be found that

ll rights reserved.

ro).

a nano-film does completely separate the contacting surfaces.Such sheared nano-films are shown to have very complicatedstatic and dynamic properties that depend essentially on themolecular structures at the solid–liquid interface [2,3]. Thetribological properties of lubricants under such confinement arethus different from those in large separations. Modern lubricantblends come with very complex chemical compositions. Undermolecular confinement, the different constituents behavedifferently according to their molecular structures [1]. Byunderstanding these different behaviours, lubricant design canimprove the predictive ability about lubrication performance atthe different tribological scales encountered in many modernapplications.

ZDDPs provide an example of classical yet very good performinganti-wear additives present in many commercial engine lubricationoil blends [4]. However, ZDDPs are also known to have undesirablehigh friction properties under confinement conditions of mixedlubrication [4]. It is suspected that interfacial effects are behind thehigh friction behaviour of ZDDPs. What is certain, however, is thatthe effect that ZDDPs have on friction is not negligible as recentexperimental studies [5] have shown that a low concentration ofC4-ZDDP in a lubricant mixture with hexadecane base oil results inabout 40% increase in friction. By using molecular dynamicssimulations with no chemical transformations, this work aims toanalyze the behaviour of C4-ZDDP in hexadecane solution undermolecular confinement conditions in the purpose of identifying themain reasons of its high friction behaviour.

In the next section, the dominant molecular phenomena inconfined liquid flow near solid surfaces are presented. Section 3details the molecular model and the simulation procedure of

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Nomenclature

k stiffness parameter for the harmonic springs betweensurface atoms (Kcal mol�1 A�2)

m atomic mass (amu)e Lennard Jones energy parameter (Kcal mol�1)s Lennard Jones distance parameter (A)q electric charge (e)jij localized interaction potential energy at position (i,j)

(Kcal mol�1)eij localized equivalent Lennard Jones energy at position

(i,j) (Kcal mol�1)sav average Lennard Jones distance parameter for an

atom of the ferric oxide surface (A)seq equivalent Lennard Jones interaction distance para-

meter between the surface and scanner atoms (A)r0 normal elevation of the scanner atoms from the

surface (A)e average surface energy (Kcal mol�1)Nx number of potential energy measurements during the

x-direction scanNy number of scanner atoms aligned across the y

dimensionC potential corrugation factor (Kcal mol�1)Ethermostat total extracted energy at the thermostat level (Kcal

mol�1)Fx sum of the tangential forces extracted on the moving

surfaces (Kcal mol�1 A�1)vx, vs surface velocity (m s�1)Pz normal loading pressure (Pa)S surface area of the molecular domain (A2)

vz normal vibration velocity of the upper surface (A fs�1)Ek total thermal kinetic energy of the lubricant (Kcal

mol�1)k molecular layer thickness (A)d mechanical slip length with respect to one surface

(nm)Lmech

s mean mechanical slip length (nm)d0 thermal slip length with respect to one surface (nm)Lthermal

s mean thermal slip length (nm)dc reduced average central distance of the ZDDP mole-

culesZi Z-coordinate of the centre of mass of ZDDP molecule

(i) (A)Zc elevation of the plane at the centre of the lubricant

mixture film (A)h film thickness during the steady state (A)v reduced average velocity of the ZDDP moleculesvi streaming velocity of the centre of mass of ZDDP

molecule (i) (m s�1)N total number of ZDDP moleculesT0 imposed temperature in the outer layer of the solid

surfaces (1K)T, Tlubricant average temperature inside the lubricant during

steady state (1K)Z lubricant dynamic viscosity (Pa s)_geff effective shear rate, average shear rate in the middle

of the film (s�1)_gapp apparent shear rate (s�1)m friction coefficienttxz apparent shear stress (Pa)

H. Berro et al. / Tribology International 43 (2010) 1811–18221812

molecularly confined lubrication by hexadecane in the presenceand absence of ZDDP. The results are presented and discussed inSection 4.

2. Background on nano-scale effects in confined lubricantfilms

It is known that liquids exhibit particular flow behaviour nearsolid surfaces [6–10]. This is mainly the result of particularlyinduced order from the organized solid surfaces on the neigh-bouring unorganized liquid [6,7]. When liquids are confined tomolecular scales, interfacial effects with the solid surfaces becomeof higher significance and the global liquid structure anddynamics are influenced in a manner that contradicts many lawsof continuum fluid dynamics [8–10]. Solid and liquid moleculareffects are dominant factors that characterize the dynamic andthermal properties of molecularly confined liquids [11,12].

The mechanical interfacial slip (velocity jump) of liquids withrespect to solid surfaces is one of the most interesting interfacialphenomena that are strongly influenced by confinement[9,10,13]. The classical continuum assumption that fluids flowat the same speed as their neighbouring solid breaks downat molecular confinements. Many experimental [14,15] andnumerical studies [10,16–18] showed that molecularly confinedliquids flow at different speeds than their neighbouring solids andhence are more weakly sheared in the bulk than expected.Consequently, the liquid flow resistance and friction can bereduced in molecularly confined systems where mechanical sliptakes place [14,19]. It remains a very intriguing phenomenon toinvestigate not only because it can be one of the few routes to low

friction in molecularly confined lubricant films but also becauseits prediction and control are not yet completely understood.

It was shown that slip, in confined systems, depends on theoperating conditions [13,16] as much as on the molecularstructure and interactions at the solid–liquid interface [9,17].Molecular dynamics simulations showed that the mechanical slipincreases with the level of forcing until a certain limit that cannotbe exceeded [16].

Mechanical slip also depends on the solid–liquid interactionstrength and the solid surfaces morphology: potential corrugation[17] and geometrical roughness [6]. Molecular simulationshave shown that the strength of the solid–liquid interaction isvery critical on the interfacial mechanical slip phenomenon.In molecular simulations, the transition from slip to no slipboundary condition was obtained by only increasing thesolid–liquid interaction strength [9,17]. As for the surfacemorphology, it was repeatedly found that for confined liquidsbetween molecularly smooth solid surfaces, velocity slip occurson the boundaries and the flow in the bulk fluid is characterizedby an almost constant shear rate [9,11,12]. Although rough and(potentially) corrugated surface structures tend to frustrate theliquid ordering in the tangential and normal directions [6,17],velocity slip was shown to be strongly reduced under theseconditions for the liquid layers adjacent to those surfaces [6].Finally, the mechanical slip also depends on the liquid molecularstructure notably on the chain length and the degree of branching.Molecular dynamics simulations showed that velocity slip waslarger for branched molecules and generally increased with thechain length [20].

Nevertheless, there have been no studies reported in theTribology literature, to the best of our knowledge, on the effect of

H. Berro et al. / Tribology International 43 (2010) 1811–1822 1813

polarity and electrostatic forces on interfacial slip of confinedlubricant films. In reality, lubricants are rarely composed of pureand simple alkanes and they usually contain a multitude ofdifferent components and additives, many of which are composedof long chains, have high degrees of branching, and may eveninclude polar sites that may interact through electrostatic forceswith the bounding surfaces.

By considering the effect of the addition of a small concentra-tion of a polar molecule (C4-ZDDP) to a saturated linear alkanechain lubricant (hexadecane) with different possibilities for solid–liquid interaction strength and surface potential corrugations, thiswork aims to contribute to the understanding of lubricantinterfacial behaviour near solid surfaces. The effect that thepresence of C4-ZDDP has on both the interfacial phenomena andfriction will be examined and the results will be compared torecent measurements of friction under mixed lubrication in thepresence of ZDDP.

Table 1Atomic masses, Lennard Jones energies (e) and distances (s), as well as atomic

charges (q) for the pair-wise interactions used in the simulations [22].

m

(amu)

e (Kcal mol�1) s (A) q (e)

Surface Fe 55.845 0.5, 1, 2.5, 7.5,

15� e0

2.20 +0.771

O 15.999 0.1699 2.96 �0.514

Hexadecane

and ZDDP

CH2 14.027 0.0933 3.93 0.000

CH3 15.035 0.2264 3.93 0.000

ZDDP O 15.999 0.1699 2.96 �0.514

P 30.974 0.1999 3.74 +1.858

S 32.974 0.2498 3.56 �0.830

Zn 65.370 0.2500 1.95 +0.600

3. Simulation details

3.1. Molecular model

Molecular dynamics simulations are set up to study the effectof solid–liquid interaction strength and the use of a lowconcentration of ZDDP additive on local friction in molecularlyconfined lubricant films. Hexadecane is used as the base oil and isstudied separately as well as with the addition of a 5% massconcentration of C4-ZDDP. The lubricant film is confined betweenmolecularly flat ferric oxide (Fe2O3) surfaces representing theportion of the bearing steel oxide layers which are in directphysical interaction with the lubricant molecules. A snapshot ofthe molecular system with the ZDDP additive is shown in Fig. 1.The initial domain size is 50�30�45 A, periodic in the shear (x)and transversal (y) directions. The z-dimension is allowed to vary(shrink wrapped) as the simulations are performed underimposed normal load and thus fluctuating film thickness.

Each oxide surface is constituted by 1455 Fe and O atoms ona rhombohedral lattice of characterized by the parameters:a¼5.029 A, c¼13.730 A and internal degrees of freedom:xO¼0.3056 and zFe¼0.10534 according to Ref. [21]. The solid–solid interactions are modelled using a spring interaction modelin which each surface atom is initially connected by unbreakable

Fig. 1. Snapshot of the molecular model simulating confined lubricant flow between

simulation box.

harmonic bonds to all neighbouring atoms within a cut-offdistance of 3 A, considering those that are initially across theperiodic boundaries. The rigidity of the solid structure dependsuniquely on the choice of the stiffness parameter k. Thisparameter was chosen as k¼130 Kcal mol�1 A�2 as a compromisebetween surface solid-like rigidity and vibration flexibility tofavour heat transfer when in interaction with the lubricantmolecules. The wall atoms interact with both hexadecane andZDDP molecules using the 12-6 Lennard Jones potentialwith a cut-off distance of 10 A and additional long-rangeColumbic electrostatic forces with no cut-off distances areconsidered for the ZDDP molecules which include non neutralatoms (Table 1). The upper and lower surfaces do not interactwith each other.

As for the confined lubricant, the initial configuration for thesimulations with pure hexadecane is prepared with 120 mole-cules of the latter, randomly distributed inside the gap betweenthe two surfaces. For the simulations with the addition of ZDDP,114 molecules of hexadecane and three molecules of ZDDP wereused. Since the volume of the simulation box is allowed to vary,the density of the lubricant is thus allowed to physically evolveaccording to the pressure and temperature. Hexadecane mole-cules are modelled using a United Atom model in which CH3 andCH2 groups are considered as single interaction sites. Hydrogenatoms are only compensated in the mass of the united atoms. The12-6 Lennard Jones potential is also used here to describe all non

Fe2O3 surfaces. The lighter shaded regions correspond to periodic images of the


bonded interactions for the hexadecane molecule. Intra-moleculararchitecture involves bond stretching, angle bending, andtorsion potentials inside each molecule. The net charge of eachinteraction site (united atoms) is zero for each hexadecanemolecule.

A schematic of the molecular structure of hexadecane andC4-ZDDP molecules modelled by united atoms is presented inFig. 2. The butyl groups of ZDDP are modelled by four unitedatoms identical to those in hexadecane whereas phosphorus,sulphur, oxygen and zinc atoms are considered explicitly. TheZDDP molecular architecture includes the same intra-molecularinteractions described for hexadecane. Also included arethe atomic charges (Table 1), from the OPLS force field [22], bywhich long-range electrostatic forces are computed in thesimulations.

3.2. Wall strength and morphology

The dynamics of confined hexadecane molecules are expectedto be very sensitive to surface morphology and interactionstrength [6,9,13]. Hexadecane molecules more easily stick topotentially corrugated and geometrically rough surfaces and slipwith respect to geometrically smooth, less corrugated surfaces.In this study, although the number density of surface atoms inthe rhombohedral lattice is elevated, the lattice orientation and

Fig. 3. The potential corrugation scanning procedure used for characte

Fig. 2. Schematic showing the molecular structures of hexadecane (top) and

C4-ZDDP (bottom) modelled with united atoms.

the presence of two atom types in the surface result in a quitecorrugated potential. Furthermore, five different choices for theLennard Jones energy parameter of the surface iron atom areconsidered (Table 1). These choices reflect different surface inter-action energies and potential corrugations. The surface energywill be considered as a key varying parameter in the simulationsof hexadecane with and without the addition of the ZDDPadditive.

In order to test the initial level of potential corrugation for eachsurface model, a surface potential energy scan procedure wasundertaken as follows: a set of 100 free scanner atoms identical tothe CH2 united atom are aligned in a straight line that extendsacross the y-dimension of the oxidized surface at a perpendiculardistance of r0¼3 A from one side of the surface (Fig. 3). The set ofatoms is then moved at constant speed and elevation scanning thepotential energy variation with 100 data points per scanner atomalong the surface x-dimension. The measured potential energy(jij) at each position can be arbitrarily converted to an equivalentLennard Jones energy (eij) felt by the CH2 scanner united atomaccording to the equation

eij ¼jij

4ffiffiffiffiffiffiffiffiffieCH2

p seq

r0

� �12

�seq

r0

� �6" #

266664

377775

2

ð1Þ

The scanner atoms’ elevation is r0¼3 A, and the Lennard Jonesequivalent distance parameter for the scanner atoms–solidsurface interactions is seq ¼

12ðsavþsCH2

Þ; sav being the averageLennard Jones distance parameter for surface atoms defined inthis case as sav ¼

15ð2sFeþ3s0Þ.

In this work, five surface models were considered. Thedifference between surfaces: (a), (b), (c), (d), and (e) correspondsto a difference in the Lennard Jones energy parameter (eFe) of theiron atoms which is chosen as 0.5, 1, 2.5, 7.5, and 15 times theenergy (eO) of oxygen atom in the surface, respectively.

In addition to an x�y snapshot of the oxide surface, Fig. 4reports the local energy eij felt by a CH2 scanner united atom atany position with respect to surface (d). It shows that thepotential energy varies across the surface due to the crystallinenature of the structure. An average surface energy e can becalculated by averaging the potential energy over the wholesurface, then determining equivalent Lennard Jones energy ofsurface atoms. This permits to quantify the equivalent effect ofsurface strength on the lubricant atoms.

e ¼PNx

i ¼ 1

PNy

j ¼ 1 jij

4ffiffiffiffiffiffiffiffiffieCH2

pNxNy

seq

r0

� �12�

seq

r0

� �6� �

2664

3775

2

ð2Þ

rizing the wall strength and corrugation of the five Fe2O3 surfaces.


e is the average surface energy, Nx the number of data pointsalong x¼100, Ny the number of data points along y¼100.

In addition to their different average energies, differentsurfaces can have different corrugations of potential. In order tocharacterize this aspect, potential corrugation (C) is computed asthe mean absolute difference of the local felt energy ðeijÞ from theaverage energy ðeÞ across the whole surface

C ¼

PNx

i ¼ 1

PNy

j ¼ 1 jeij�ejNxNy

ð3Þ

The surface scan procedures described in this section with theintroduction of e and C allow the surface potential corrugationand mean strength to be quantified. In the simulations carried outfor this study, surface (e) is shown to have the highest energy as

Fig. 4. x�y snapshot of the ferric oxide surface (top) and potential energy scan

results for surface (d) used in simulation A-4 (bottom). The black and white atoms

represent Fe and O atoms, respectively.

Fig. 5. Surface strength and potential corrugation according to the choice of the

Lennard Jones energy parameter for iron atoms. Simulations ‘‘A’’ correspond to

lubrication by pure hexadecane whereas simulations ‘‘B’’ consider the addition of

5% of ZDDP to hexadecane.

well as the highest potential corrugation. Surface (d) has a higherenergy and corrugation than (a), (b), and (c) and so on as shownin Fig. 5. Since e and C, representing the surface strength andphysical roughness, are supposed to be of significant importancethey will be considered as key varying parameters in thefollowing.

3.3. Simulation procedure

In order to identify the effect of the ZDDP additive, two sets ofsimulations are considered. In simulations ‘‘A’’, the lubricant iscomposed of only hexadecane molecules whereas in simulations‘‘B’’, the lubricant is a mixture of hexadecane and ZDDP (5% massconcentration). In all simulations a uniformly distributed load of1.3 GPa, equivalent to the experimental conditions of Ref. [5], isapplied by adding equal elementary normal forces on all atomsinside one 2A-thick layer of each of the upper and lower surfaces.These layers are chosen furthest from the solid–liquid interface asshown in Fig. 1. The elementary forces being of oppositedirections for the upper and lower surfaces, they result in thecompression of the confined lubricant. The sum of these forces ineach surface divided by the surface area is 1.3 GPa.

Each simulation consists of three main phases: initialization,compression, and shear. In the initialization phase, we proceed byenergy and force minimization procedure by the CG (ConjugateGradient) algorithm. The positions of lubricant atoms are slightlymodified from the initial disposition minimizing the totalpotential energy and the interaction forces in the system. Byfreezing the two surfaces, the lubricant medium is then relaxedduring 500,000 time steps, the equivalent of 0.5 ns of simulationtime. In the second phase, the uniformly distributed loads areapplied to both the upper and lower surfaces thus compressingthe lubricant in between. At this stage, we apply a global Nosethermostat to the system keeping it at a temperature of 300 K.The temperature damping coefficient for the thermostat is 50 fswhich signifies that temperature is relaxed in a time span of 50 fs.With constant load, the film thickness and thus the density ofthe lubricant vary in a damped oscillatory manner during thisphase. A constant film thickness is attained in about 0.2 ns butthe compression is continued for a total of 0.5 ns to insure theequilibrium state of the lubricant.

At the end of the compression phase, the surfaces are movedwith equal but opposite constant velocities of 10 m s�1 on thex-direction resulting thus in lubricant shear. During this phase,energy is dissipated using a localized boundary Nose thermostatof 300 K with identical damping properties as the one used in thecompression phase. This thermostat acts on a 2A-thick layer ofeach surface as shown in Fig. 1. At the shear onset, we observe atransient response inside the film, notably an expansion due tothe increase of temperature by shear heating. The steady state isattained in all simulations after 0.3 ns from the shear onset. Thesampling process starts 0.5 ns after the onset of shear, insuringthus that the shearing steady state is attained and is continued for2.5 ns. The total simulation time including all of the mentionedphases is 4 ns, including 2.5 ns of sampling.

The time step used in the simulations is 1 fs¼10�15 s. TheVelocity-Verlet algorithm is used for numerical integration ofthe atomic classical equations of motion. Finally, long-rangeelectrostatic forces for simulations ‘‘B’’ with ZDDP were calculatedusing a particle–particle particle-mesh solver [23]. This solverallowed the simulations to be completed about eight times fasterthan explicit Ewald summations for electrostatic forces.

Five simulations were conducted using pure hexadecaneas lubricant (group A) with five different types of surfaces(as previously characterized with e and C) whereas three


simulations were run with the addition of the ZDDP additive(group B). Fig. 5 presents the surface properties (strength andcorrugation) used in each of the eight simulations.

3.4. Energetic balance validation

Molecular dynamics simulations generate a nanoscopicevolution of the molecular system. For a system of N atoms, thisevolution represents the trajectory in an (N�N) phase space ofatomic positions and momenta. Some processing with statisticalmechanics laws is thus always necessary to derive macroscopicscale information of interest. However, the derived macroscopicinformation may have no physical meaning, comparable toexperiments, unless the system is in a certain state of thermo-dynamic equilibrium. In other words, it is necessary to verify thatthe instantaneous microscopic state of the system during itsevolution in phase space belongs to a well defined thermo-dynamic ensemble.

In non-equilibrium shear simulations as described in thiswork, the lubricant thermodynamic state needs to be identifiedbefore calculating any macroscopic variables of interest. It isimportant to emphasize that in all simulations in this work, noconstraints are imposed directly inside the lubricant; only theindirect effects transmitted from the surfaces by conservativeforces are present. At the shear onset, the thermodynamic state ofthe lubricant film is indeed very complicated and distinct fromany definition of global or local equilibrium. When the surfacesbegin to move, their momentum difference diffuses graduallyinside the lubricant film through the molecular interactions andthe lubricant film is sheared. The lubricant shear, then, gives riseto an increase in thermal kinetic energy and consequently thelubricant expands. However, this transition phase occurs duringa fraction of a nanosecond, and then the lubricant kinetic andpotential energies become constant. This is referred to as thenon-equilibrium steady state. At this state, local thermodynamicequilibrium is found inside the formed molecular layers and itbecomes convenient to define macroscopic variable fields in eachlayer such as velocities and temperatures.

The constant energy of the lubricant is related to the energeticbalance which develops at the non-equilibrium steady stateas shown in Fig. 6. The power of the total external mechanicalforces (tangential and normal mechanical power) equilibrateswith the extracted heat energy by the thermostats at the surfaceslevel. Hence, from an energetic point of view during the

Fig. 6. Energetic balance and film thickness stabilization during the shearing

steady state of simulation A-1 with pure hexadecane. The apparent shear rate is

6.87�109 s�1.

non-equilibrium steady state, the lubricant becomes aconservative energetic path between the upper and lowersurfaces where kinetic and thermal diffusions occur. Thedynamic diffusion is due to the momentum difference betweenthe upper and lower surfaces whereas the thermal diffusionrepresents heat extraction from the lubricant toward thethermostated surfaces. Since the lubricant volume (filmthickness) (Fig. 6) and total number of atoms are also constantduring the steady state, the phase space trajectory of the lubricantmolecular system belongs to an equilibrium NVE thermodynamicensemble.

In what follows, the results for the interfacial flow propertiesand tribological performance of the lubricants are presented anddiscussed for simulations ‘‘A’’ and ‘‘B’’ in two separate sections.The discussion on the results of simulations ‘‘B’’ highlights thedifferences observed through the addition of the ZDDP additiveunder confinement conditions. The reader is advised to refer tothe Appendix A for details about the computation methods ofvariables and profiles on which the following discussions arebased.

4. Results and discussion

4.1. Simulations ‘‘A’’: pure hexadecane

A hexadecane film (initially of 3 nm thickness) is confinedbetween two ferric oxide surfaces. The two surfaces are movedwith constant but opposite velocities resulting in the shearingof the hexadecane film (see Section 3 for details). Five types ofsurfaces are considered by varying the Lennard Jones coefficientof the iron atoms in the surfaces as explained in Section 3. Thesesimulations thus aim to quantify the effect of surface potentialstrength and potential corrugation on a neighbouring hexadecaneflow.

4.1.1. Mechanical slip

Fig. 7 presents the velocity fields across the hexadecanelubricant film for different wall energies. Each point correspondsto the average flow velocity inside one layer of the lubricant filmas described in the Appendix A. The z-dimension is thus defined interms of the molecular layer thickness k which was found to beequal to 4.2 A in all simulations. It is seen that the lubricant slipsfor the cases with low equivalent Lennard Jones wall energy(simulations A-1 and A-2). In this case, the internal lubricant

Fig. 7. Velocity profiles across the hexadecane lubricant film for each surface type.


cohesion is stronger than the cohesion with the walls and thusshearing is easier at the interface than it is inside the lubricant.As a result, the effective lubricant shear rate is smaller than theapparent one as shown in Fig. 8a. A more quantitative repre-sentation of mechanical slip is given in Fig. 8b with the mecha-nical slip length as a function of the wall energy. The lubricantslips when measured mechanical slip length is positive.

By increasing the wall energy, the slip condition is graduallysuppressed. Globally, the fluid no longer slips when the calculatedvalue of slip length becomes zero or attains apparently negativevalues (Fig. 8b). The apparently negative values of the slip lengthare due to lubricant layers completely sticking to the solidsurfaces which occur in the cases with elevated wall energies andcorrugation (simulations A-4 and A-5). The lubricant layers atthe solid interface are said to be ‘‘locked’’ to the walls. In this case,the actual shear starts from a distance inside the lubricant and notfrom the surfaces level. In consequence, the average shear rate ofthe fluid is larger than the apparent one (Fig. 8a).

4.1.2. Thermal slip

Thermal slip is related to the temperature jump at thesolid–liquid interface (see Appendix A). It characterizes theinterfacial thermal resistance which is also known as the Kapitzaresistance [12]. In the case of confined lubricants under shear, thegenerated heat is transmitted through the solid–liquid interface

Fig. 8. Variation of the apparent and effective shear rates (Fig. 8a) and the mean mec

effective shear rate—filled circles and boxes; apparent shear rate—empty circles and b

Fig. 9. Variation of the lubricant mean temperature (Fig. 9a) and the ther

to the solid surfaces where it is eventually dissipated. If theinterfacial resistance is high, the generated heat is slowlydissipated across the solid–liquid interface and results in a highertemperature difference between the solid and lubricant. Theresults for the mean temperature inside the confined hexadecanefilm are presented in Fig. 9a. The results indicate thatthe lubricant temperature increased from its initial state, andthe continuously imposed temperature of 300 K at 6 A in the soliddepth, by around 20–30 K. This phenomenon is due to thelubricant heating at the elevated shear rates encountered inthese simulations.

The inspection of temperature profiles of Fig. 10 shows that thelubricant heating also influences the solid surfaces and results in asmall temperature gradient at the surfaces level. However, thereis a remarkable discontinuity in the temperature profiles atthe solid–liquid interface especially for the simulation with thelowest wall energy (A-1). The temperature jump at solid–liquidinterfaces is related to the Kapitza interfacial thermal resistanceand is characterized by the thermal slip length as shown inFig. 8b. The results of thermal slip length show that it alsogenerally decreases with the wall energy in analogy withthe mechanical slip length. However, thermal slip still occurs inthe absence of dynamic slip (simulations A-3, A-4, and A-5).

The relationship between the lubricant temperature at thesteady state and the wall energy is quite complex. It is due tothe counteracting effects of two phenomena: heat generation by

hanical slip length (Fig. 8b) as a function of the equivalent wall energy. (Fig. 8a:

oxes).

mal slip length (Fig. 9b) as a function of the equivalent wall energy.

Fig. 10. Temperature profiles across the hexadecane lubricant film and in the

neighbouring solid layers.

Fig. 11. Variation of the friction coefficient as a function of the equivalent wall

energy.


lubricant shear and interfacial thermal resistance. When the wallenergy is increased, the lubricant is sheared with a highereffective shear rate due to the decrease in mechanical slip(Fig. 8a). Thus, more heat is generated inside the lubricants insimulations with higher wall energies. However, this does notnecessarily mean that the final temperature will be higher forthese cases. The final lubricant temperature depends on anotherfactor which is the resistance in the extraction path of thisgenerated heat. In the conducted simulations, an outer layer ofeach surface is thermostated, thus the generated heat istransferred from the lubricant to the surfaces, through thesolid–liquid interface, and then it is conducted inside the solidsurface into the thermostat level (or to the surroundings inreality) where it is permanently removed.

The increase in wall energy has a direct effect on the interfacialthermal resistance which was characterized by the thermal sliplength in Fig. 9b. It results in a general decrease in the thermal slipand hence in a decrease in the interfacial thermal resistance.As a result, the lubricant can more easily pass the generated heatto the thermostat in the cases of higher wall energies.

The results of Fig. 8a show that by increasing the wall energy,although more heat is generated inside the lubricant film due tothe increase in the effective shear rate, this heat is more easilytransferred across the solid–liquid interface and eventuallyremoved by the thermostat. Hence, the mean lubricant tempera-ture at the steady state is generally lower for higher wall energies.

4.1.3. Friction coefficient

The results on the complex interfacial phenomena for differentwall energies have shown direct effects on the global conditionsacting on the confined lubricant film, notably the effective shearand the mean steady state temperature. In a more global point ofview, although the lubricant flow resistance is an internal fluidproperty, the change in the global acting conditions of shear rateand temperature influences the flow resistance. The effectiveshear viscosity is thus a function of the effective lubricant shearrate, the mean temperature, in addition to the loading pressure

Z¼ Zð _geff ; T; PzÞ ð4Þ

_geff is the effective shear rate. It depends on the apparent shearrate and the mechanical interfacial slip

_geff ¼ _geff ð _gapp; Lmechs Þ ð5Þ

The lubricant temperature as explained previously dependsalso on the effective shear rate which relates to the generated heat

as well as the thermal slip length that characterizes the interfacialthermal resistance

T ¼ Tð _geff ; Lthermals Þ ð6Þ

The friction coefficient is finally the ratio of the apparent shearstress txz divided by the normal pressure Pz. The apparentlubricant shear stress can be written as the effective shearviscosity Z multiplied by the effective shear rate _geff as shown inEq. (7):

m¼ txz

Pz¼

1

PzðZ � _geff Þ ð7Þ

Eqs. (4)–(7) show that the friction coefficient depends on theinterfacial behaviour of the lubricant film (Lmech

s and Lthermals ) in

addition to the operating conditions ( _gapp and Pz). Since themechanical and thermal slip lengths were shown to be related tothe wall energy (Figs. 10a and 11b), it becomes evident that thefriction coefficient will also depend on this factor.

The increase of wall energy results in an increase of theeffective shear rate (Fig. 8a) as well as a slight decrease ofthe lubricant steady state temperature (Fig. 9a). In this context,the shear rate—and temperature—viscosity dependences (Eq. (4))oppose each other. The increase of wall energy hence does nothave a major effect on the lubricant shear viscosity. The majoreffect of the wall energy on friction remains through theproportionality between friction and the effective shear rate(Eq. (7)). As the effective shear rate increases with wall energy(Fig. 8a), the friction coefficient also increases as shown in Fig. 11.

4.2. Simulations ‘‘B’’: lubricant mixture

The simulations in this section differ from the preceding onesby the addition of 5% mass concentration of the ZDDP additivemolecules initially placed in the centre plane of the gap betweenthe two surfaces. Three types of surfaces were examined in thissection representing surfaces (a), (c), and (e), described previously(see Fig. 5).

4.2.1. Molecular organization

The time evolution of the reduced average centre distance dc

and the reduced average velocity v (both described in theAppendix A) of the ZDDP molecules in simulation B-1 (weak wallenergy) are presented in Fig. 12. At the beginning of shear, dc isnear 0 which indicates that ZDDP molecules are present in themiddle of the lubricant film. However, dc increases during the first

Fig. 12. Time evolution during simulation B-1 of: (1) the reduced average distance

dc of the ZDDP molecules from the centre of the film, and (2) the reduced average

velocity v of the ZDDP molecules with respect to the surface velocity vs.

Fig. 13. Velocity profiles across the hexadecane+ZDDP lubricant mixture film for

each of the three surface types.


1 ns and reaches a maximum value of about 0.9. This indicatesthat the ZDDP molecules migrate from the centre of the lubricantfilm toward the surfaces and then are completely disposed onthe surfaces. The value of dc does not reach 1 because of the gapsbetween the faces of each of the two surfaces and the firstlubricant layer. This gap is due to the repulsive part ofthe potential that prevents any molecules from penetrating thesurfaces.

The results for the average velocity v of ZDDP are very similar,and show the variation in the velocity of the ZDDP moleculesduring their migration process. It is of importance to remark thatthe ZDDP molecules stick to the surfaces even in the case of weakwall energy (simulation B-1) and this is characterized by the valuev ¼ 1. For the same wall energy and with pure hexadecane,a significant wall slip was observed (Fig. 7). Similar migrationbehaviour is observed for the other two simulations B-2 and B-3.

4.2.2. Mechanical slip

The previous result for ZDDP average velocity and the stickingeffect of ZDDP molecules with respect to the surfaces has a directeffect on the lubricant mixture velocity profile given in Fig. 13.Contrary to the results obtained with pure hexadecane, thelubricant mixture satisfies a no slip boundary condition; even atthe lowest wall energy used in simulation B-1, the lubricant filmis locked to the surfaces. Since the hexadecane moleculesare supposed to slip for this wall energy, the result proves thatthe presence of only 5% ZDDP inside the film can suppress themechanical slip phenomenon due to the additive migration andsticking to the surfaces. Thus, ZDDP molecules act like additionalsurface roughness that entrains the whole lubricant layer at thewall speed.

The quantitative examination of the results for slip lengthversus wall energy (Fig. 8b) confirms these observations. Theaddition of only 5% mass concentration of ZDDP thus results in thedisappearance of velocity slip with respect to the walls. Inconsequence, the effective shear rate in the centre of the lubricantis higher than the apparent shear rate as the actual shear startsfrom a distance inside the lubricant and not from the surfaceslevel (Fig. 8a). The effect of increasing the wall energy onmechanical slip is similar to the simple hexadecane case as itresults in an ‘‘apparently more negative’’ slip length and thus anincrease in the effective shear rate.

4.2.3. Thermal slip

The results for the thermal slip length in the mixturesimulations are reported in Fig. 9b. Although the mechanical slipwas suppressed by adding the ZDDP additive, there was littleeffect on the thermal slip. In the case of weak wall energy(simulation B-1), the addition of ZDDP reduced the thermal sliplength and thus the interfacial thermal resistance by improvingthe molecular adherence and organization near the walls.However in the two other cases (A-3 and B-2 then A-5 and B-3)where no velocity slip is seen for pure hexadecane, the presenceof ZDDP results in an increase in the interfacial resistance that isdue in this case to the heterogeneity in the molecular structure,due to ZDDP migration as illustrated in Fig. 12, of the lubricantlayer adjacent to the surfaces which frustrates the thermaltransfer.

The increase of wall energy results in simulations ‘‘B’’ in adecrease in the thermal slip length. However, the thermal slip waspresent for all cases even though no mechanical slip occurs. Thisresult is important because it shows that the thermal slip mayoccur even for the cases where no mechanical slip is seen. It canbe concluded hence that the mechanisms responsible for dynamicand thermal slips are not the same.

Furthermore, the presence of ZDDP results in an increase of themean lubricant temperature which is due to both the increase inthe heat generation (due to the increase in the effective shearrate) and the decrease in the interfacial thermal resistancecompared to the hexadecane simulations. The mean lubricanttemperature is a decreasing function of the wall energy due to theeffect that the wall energy has on the thermal slip (Fig. 9b). Higherwall energy results in a smaller thermal slip length and thuseasier transferability of the generated heat in the contact.

4.2.4. Friction coefficient

The results for the friction coefficient are presented in Fig. 11as a function of the wall energy. The presence of ZDDP results inan increase in friction as shown in experiments [5]. The effects ofthe molecular migration of ZDDP toward the surfaces on theinterfacial flow result in an increase of the effective shear rate inthe middle of the film.

As for the influence on the flow resistance, the presence ofZDDP resulted in an increase in the lubricant temperature(Fig. 9a). In this case, heating effects tend to decrease theshear viscosity of the lubricant in the ZDDP case (B) compared tothe hexadecane case (A). However, this has little effect on thefriction coefficient and the major factor that contributes to


friction remains the effective shear rate that the lubricant film issubject to.

It is remarkable that the friction coefficient in the hexadecane+ZDDP mixture case (B) is less sensitive to wall energy comparedto the hexadecane case (A). The reason is that the passage from lowto high wall energy had no effect on the no slip boundary conditionin the presence of ZDDP. This is in contrary with the hexadecanecase (A) where the change in the wall energy had consequenteffects on the mechanical slip boundary condition.

It is finally convenient to perform a direct comparison aboutfriction in simulations A-5 (hexadecane) and B-1 (hexadecane+ZDDP). Although very different surface types are considered inthe two simulations (Fig. 5), the interfacial behaviour from thetwo simulations is very similar and is illustrated by very closeeffective shear rates (Fig. 8a). In simulation A-5, the no slipboundary condition was due to the strength and corrugation ofthe wall potential. In simulation B-1, the no slip boundarycondition was uniquely due to the presence of the ZDDP additivesince the surface with the weakest wall energy and corrugationwas considered.

The temperature increase in the two simulations shown inFig. 9a, however, differs. As illustrated previously, the lubricanttemperature is due to both the heating phenomenon and theinterfacial thermal resistance that works against the possibilityfor heat extraction. Although the generated heat is supposed to bethe same for the two simulations since the effective shear ratesare equal, the results in thermal slip length between the twosimulations (A-5 and B-1) shown in Fig. 9b are very different. Ahigher thermal slip is reported for simulation B-1 (Fig. 9b) whichmeans that the interfacial thermal resistance in this case is higher.In consequence, the same generated heat, in each of the twosimulations, is extracted with more difficulty in the simulationwith ZDDP (B-1) and the result is a higher temperature inside thelubricant mixture for simulation B-1 as shown in Fig. 9a.

Finally, in terms of friction, Eq. (7) relates the discussed factorsaltogether in the calculated friction coefficient. In both simula-tions, the intermediate lubricant was subjected to the sameeffective shear rate. The difference in the apparent shear stressesis thus the result of thermal effects, notably the temperaturedependency of shear viscosity. The lower lubricant temperature insimulation A-5 (Fig. 9a) plays a role in increasing the shearviscosity and results in a higher shear stress for the same shearrate of simulation B-1.

5. Conclusion

The presented work aimed to study the influence of differentinterfacial phenomena on friction in molecularly confinedhexadecane films. The interfacial effects were investigated for avariety of surface models with different energies and potentialcorrugations. The effect of ZDDP, a well-known anti-wear additivewas also examined. Various results related to both mechanicaland thermal interfacial phenomena were reported and consideredin the discussion.

It was shown that one route to low friction can be byconsidering surfaces with low energies and potential corruga-tions. The lubricant slip at the surface level results in loweraverage shear rates, lower stress, and thus lower friction. When asmall concentration of ZDDP additive is added, the lubricant slipcan be completely suppressed to the limit of lubricant locking atthe surface level, resulting in a higher lubricant shear and higherfriction. This result is coherent with a recent experimental studyunder similar conditions.

The interfacial thermal resistance was shown to decreasewith the wall energy but was never zero for the studied cases.

Although ZDDP decreases this resistance for very ‘‘slippery’’ sur-faces, the presence of ZDDP increases the interfacial resistance fornon-slippery surfaces due to the heterogeneity of the molecularcomposition of the lubricant layer adjacent to the walls.

Lubricant heating and interfacial thermal resistance wereshown to have an effect on the steady state lubricant temperature,a very important factor that influences the lubricant shearviscosity. The relationship between the generated heat by shearand mechanical slip length on one hand, and the relationshipbetween the thermal resistance and the thermal slip length(Kapitza length) were illustrated. It was shown that the lowthermal resistance with high wall energies allows the generatedheat to be efficiently extracted resulting in a lower lubricanttemperature during the steady state.

Finally, the previous effects were all considered to conclude onthe global effect of the localized interfacial phenomena onfriction. It was shown that the first factor that influences frictionin molecularly confined lubricant films is the effective shear ratein the middle part of the film; this depends on both the apparentshear rate and the mechanical slip length. On a second level, theinterfacial thermal resistance influences the lubricant tempera-ture increase, its shear viscosity and thus its flow resistance.

Acknowledgements

This work is part of H. Berro’s Ph.D. thesis which is funded bythe French MENESR (French Ministry of Higher Education andScientific Research) on a 3-year scholarship basis. MolecularDynamics simulations were possible thanks to the open sourcemultipurpose molecular dynamics simulator ‘‘LAMMPS’’ bySandia National Laboratories, USA. The necessary computationalresources were possible through LaMCoS computing facilities inaddition to the generous P2CHPD cluster, member of FLCHP(Lyon’s Federation of High Performance Calculation).

Appendix A. Variables and profiles computation methods

The local state of the lubricant is characterized by its density,velocity, and temperature profile. In the simulations with ZDDPadditive, the distance from the centre of the film and thevelocities of ZDDP molecules are also determined to characterizeany local phenomena due to the additive in the lubricant mixture.Finally, the average shear rate, mechanical and thermal sliplengths, and the friction coefficient are measured in order todefine the global state of the lubricant. The methodology ofcalculation of all of these scalar variables and profiles are definedin this section.

Lubricant mass density profile: The confined volume is cut,along the film thickness direction, into thin layers of 1 A thicknessparallel to the shear direction. At each time step and in each layer,the masses of all lubricant atoms (and united atoms) that liewithin the layer are summed up and then divided by theelementary volume of each layer (50�30�0.1 A3) giving theinstantaneous mass density profile (Fig. 14). This profile is thenaveraged over 2.5 million time steps after the shearing steadystate corresponding to 2.5 ns of simulation time.

Lubricant velocity profile: The molecular organization oflubricant molecules in layers parallel to the shear direction is awell known phenomenon in confined liquids between solidsurfaces [24–27]. The order of the solid surfaces is influenced onthe liquid structure especially near the solid–liquid interface. Thiseffect fades away gradually further from the surfaces as shown inFig. 14. In a previous work [28], local thermodynamic equilibriumwas shown to exist inside these molecular layers. Moreover, it


was shown that proper kinetic energy decomposition into shearflow and peculiar thermal components for each layer can beobtained from flow velocities calculated as the layer total linearmomentum divided by the layer total mass, rather than theconventional methods by simple averaging of atomic velocities bythe number of atoms [28].

Lubricant temperature profile: The Hybrid Diffusion method, asdescribed in a previous work [28], is used here to decomposeatomic velocities into shear flow and peculiar thermal terms ineach molecular layer, the latter being used to calculate the kineticlayer temperatures. The time average of the kinetic temperatureduring the steady state is reported in each layer to yield thelubricant temperature profile.

Shear stress/shear rates: The apparent shear stress is deter-mined as the sum of all elementary forces in the shear directionwhich oppose the constant speed motion of the two surface layersdivided by the surface area. The apparent shear rate is given as therelative velocity of one of the solid surfaces with respect to theother divided by the average film thickness during the steadystate. Finally, the effective shear rate is determined as the slope ofthe linear fit considering the points in the lubricant velocityprofile corresponding to all layers except the two that areadjacent to the surfaces.

Fig. 15. Calculation method of: Lmechs , the mean mechanical slip lengt

Fig. 14. Mass density profile across the hexadecane film during the shear steady

state of simulation A-3.

Mean mechanical slip length: The lubricant velocity profilecorresponding to all layers except the two that are adjacent to thesurfaces is fitted to a linear regression function. As shown in theschematic of Fig. 15a, the points of intersection PL and PU betweenthis fitted line and vertical dotted lines representing the surfacevelocities were determined. The mechanical slip lengths (d1 andd2) on each side are then calculated as the z-penetrations of thesepoints inside the respective surfaces level. The ‘‘surfaces level’’ isdefined at a middle distance from the inner surface face (lubricantside) and the adjacent lubricant layer. Apparently negative sliplengths correspond to cases where the points of intersection (PL

and PU) lie outside the surfaces. The absolute value of the negativeslip lengths correspond thus to the average thickness of the fluidlayers that are locked to the surfaces. Finally, the reported meanmechanical slip length Lmech

s is considered as the average of theupper surface and lower surface mechanical slip lengths.

Mean thermal slip length: The procedure to estimate thethermal slip length is illustrated in the schematic of Fig. 15b. Incomparison with the mechanical slip length, the thermal sliplength is due to a discontinuity in the temperature profilewhereas the mechanical slip length is due to a discontinuity inthe velocity profile. Since only a single layer of each solid surfaceis thermostated at temperature T0¼300 K, a temperature gradientdevelops inside the solid surfaces and the surface temperature Ts

becomes higher than T0 due to the effect of lubricant heating. Thelubricant temperature profile is linearly extrapolated from eachside of the surfaces as shown in Fig. 15b. The penetrations of thepoints of intersection with the surface temperature define thethermal slip lengths on each side (d01 and d02). The mean thermalslip length is finally given as the average of the thermal sliplengths on each side.

Friction coefficient: The friction coefficient is calculated as theratio of the total tangential shear stress (as indicated previously)to the total normal pressure applied on the upper surface.

Reduced average centre distance of ZDDP molecules: In order tofollow the motion of the ZDDP molecules during the simulations,the reduced average distance dc of the centre of masses of theZDDP molecules with respect to the elevation of the centre of thelubricant film is calculated as follows:

dc ¼

PNi ¼ 1 jZi�Zcj

Nðh=2Þð8Þ

N being the total number of ZDDP molecules in the molecularsystem, Zi the elevation of the centre of mass of ZDDP molecule ‘‘i’’,Zc the elevation of the centre of the lubricant film during steady state

h (Fig. 15a), and Lthermals , the mean thermal slip length (Fig. 15b).


and finally h the film thickness during the steady state. A value of dc

near 0 indicates the presence of the majority of the ZDDP moleculesin the centre of the film whereas a value of dc near 1 indicates thepresence of the ZDDP molecules at the surface proximity.

Reduced average velocity of ZDDP molecules: Similar to thereduced average distance dc , the reduced average velocity v ofthe centre of masses of the ZDDP molecules with respect to thesurface velocity vs is calculated as follows:

v ¼

PNi ¼ 1 jvi=vsj

Nð9Þ

N being the total number of ZDDP molecules in the molecularsystem, vi the velocity of the centre of mass of ZDDP molecule ‘‘i’’,and finally vs the surface velocity (fixed at 10 m s�1). A value of v

near 0 indicates that the shear flow velocity of the majority ofZDDP molecules is around 0, which is probable in the centre of thefilm. A value of v near 1 indicates that the majority ZDDPmolecules are moving at same velocity as the surfaces and thuscan be said to be sticking to the surfaces.

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