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Relaxation of surface tension in the liquid-solid interfaces of Lennard-Jones liquids

A.V. Lukyanov, A.E. LikhtmanSchool of Mathematical and Physical Sciences, University of Reading, Reading RG6 6AX, UK

We have established the surface tension relaxation time in the liquid-solid interfaces of Lennard-Jones (LJ) liquids by means of direct measurements in molecular dynamics (MD) simulations. Themain result is that the relaxation time is found to be weakly dependent on the molecular structuresused in our study and lies in such a range that in slow hydrodynamic motion the interfaces are

expected to be at equilibrium. The implications of our results for the modelling of dynamic wettingprocesses and interpretation of dynamic contact angle data are discussed.

The wetting of solid materials by a liquid is at the heartof many industrial processes and natural phenomena.The main difficulty in theoretical description and mod-elling of wetting processes is the formulation of boundaryconditions at the moving contact line [13]. For exam-ple, the standard no-slip boundary condition of classi-cal hydrodynamics had to be relaxed to eliminate thewell-known non-integrable stress singularity at the con-tact line [15].

The principal parameter of the theoretical descriptionis the dynamic contact angle, which is one of the bound-ary conditions to determine the shape of the free surface[13]. The notion of the contact angle has two mean-ings in macroscopic modelling. One is apparent contactangle a, which is observed experimentally at some dis-tance from the contact line defined by the resolution ofexperimental techniques (usually about a few m) andanother one is true contact angle right at the contactline. When the contact line is moving, the apparent con-tact angle deviates from its static values and becomes afunction of velocity. For example, quite often the contact-angle-velocity dependencea(U) observed in experimentscan be accurately described by

cos a = cos 0 a1sinh1(a2U), (1)

wherea1, a2 are material parameters depending on tem-perature and properties of the liquid-solid combination,U is the contact-line velocity and 0 is the static con-tact angle[3]. However useful relationship (1) may be, itis neither general, due to the well known effects of non-locality[6,7], nor it can be directly used in macroscopicmodelling since it is the true contact angle which entersthe boundary conditions used in macroscopic analysis.While the apparent contact angle can be experimentallyobserved, the true contact angle can be only inferred fromtheoretical considerations or from microscopic modellingsuch as MD simulations. This is the one of the main fun-damental problems of wetting hydrodynamics, and thatproblem, despite decades of research, is still far from acomplete understanding. The main question still remainsopen and debates continue: how (and why) does the truedynamic contact angle change with the contact-line ve-locity?

The simple hypothesis that = 0 has been usedin the so-called hydrodynamic theories, for example [8],where the experimentally observed changes in the appar-

ent contact angle were attributed to viscous bending ofthe free surface in a mesoscopic region near the contactline. Some early observations of the meniscus shapes atthe contact line have indicated that indeed the menis-cus curvature may strongly increase at the contact line[9]. The subsequent analysis has shown that while thedynamic contact angle effect may be purely apparent insome cases, it was difficult to rule out variations in thetrue contact angle. Later on, a numerical study of slipmodels has shown that whereas viscous bending can con-tribute to the observed changes in the apparent contactangle, this effect alone is insufficient to explain observa-tions[10]. Moreover, recent MD simulations of spreadingof LJ liquid drops have shown that the true contact an-gle does change with the velocity and produce a contact-line-velocity dependence similar to (1), [1113]. The re-sults of MD simulations have been successfully comparedagainst the molecular-kinetic theory (MKT) [11, 12]. Inthe MKT, which is also in a good agreement with ex-periments [3], the true contact angle is a function ofvelocity. This velocity dependence comes from the dif-ference in the probability (asymmetry) of molecular dis-placements parallel to the solid substrate at the moving

contact line, according to the phenomenological assump-tions made in the model. The asymmetry is proportionalto the contact-line velocity and, on the other hand, tothe work done by a macroscopic out-of-balance surfacetension forcefc = LV(cos 0 cos ) acting on the con-tact line. The net result is (1) with a1 = 2kBT /LV

2,a2 = (2k0)1, whereLVis surface tension at the liquid-gas interface,kBis the Boltzmann constant,Tis the tem-perature and k0 is the frequency of displacements overthe distance , which is regarded as the inverse relax-ation time of the surface phase (k0)1 = LS which issupposed to be proportional to the viscosity LS .The key feature of the model is the concentrated force fc

acting on the contact line, similar to the resistive forceintroduced in [14], so that the MKT is local.

Since the MKT is local, it would be difficult to ex-plain effects of non-locality solely within the model. Amore general and potentially universal approach to mod-elling the dynamic wetting, the interface formation the-ory, has been proposed by Shikhmurzaev [1,2]. The self-consistent macroscopic approach naturally introduces dy-namic contact angle through dynamic values of surfacetension on forming liquid-solid interfaces. The approach

arXiv:1310.5658v

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FIG. 1: Profiles of the integrand of (2) in static (at 12 =0.9, LS(5) = 0.89, 0 = 15

, and 12 = 0.37, LS(5) =0.68, 0 = 138

, in the plain geometry) and dynamic (sluggeometry, Fig. 4,averaged over za = 10 at the contact line,U = 0.1 and 12 = 0.9) conditions for a liquid with NB = 5at T = 0.8. The dashed line shows surface tension level at0 = 138

.

is very appealing and has shown excellent agreement withexperimental observations [15], but requires the knowl-edge of macroscopic surface tension relaxation time LSof the liquid-solid interface which is also supposed to beproportional to viscosity, LS 4 106 Pa1, [15].As a consequence, the theory is truly non-local, thatis able to explain effects of non-locality [6, 7, 16, 17],with the key characteristic feature, the relaxation tailof the dynamic surface tension with the length scale U LS(CaSc)1, Ca = U/LV, Sc 5(LS/6109 s)1/2(1.5

103Pa s/)1/2 is a non-dimensional ma-

terial parameter defining the strength of the interface for-mation effect cos 0 cos CaSc[15].

In summary, we have at least two principally differentmodels of dynamic wetting, both of them seem to be ina very good agreement with experimental data, [3, 15].But, which mechanism does actually determine the dy-namic contact angle? To what extent the dynamic inter-faces can be in non-equilibrium conditions and contributeinto the dynamic contact angle effect?

The key to answering those questions appears to be thesurface tension relaxation timeLSof the liquid-solid in-terface, and in this Letter, we directly establish this fun-damental parameter by MD simulations. The simulations

have been conducted in a model system consisting of LJparticles and/or chain molecules. We investigateLSde-pendence on liquid viscosity and temperature, and con-duct direct MD experiments with dynamic contact angleto get insights into the mechanism of dynamic wetting.

The MD model we use is similar to [18] but with the

LJ potentials ijLJ(r) = 4ij

ijr

12 ijr 6

and the

cut off distance 2.5 ij . Here i and j are either 1 or2 to distinguish between liquid and solid wall particleswith the masses mi. Note, hereafter, all units are non-

FIG. 2: Relaxation of liquid-solid surface tension (integratedtoym = 5) at different temperatures Tand molecular compo-sitionsNB after switching the interaction parameter 12 from0.2 to 0.65 at t0 = 100 during ts = 1. The data are aver-aged over ta = 1 and 200 independent experiments. Thedashed line is fit f0. The inset shows individual dependenceatT = 0.8 andNB = 5. The solid (red) line in the inset is fit

f1.

dimensional, the length is measured in 11, energy andtemperature in11, mass inm1and time in11

m1/11.

The beads interacting via LJ potentials are connectedinto linear chains ofNB beads by the finitely extensiblenon-linear elastic (FENE) springs, and the strength ofthe springs is adjusted so that the chains cannot cross

each other, FENE(x) = k2R20ln

1 xR0

2. Here

R0 = 1.5 is the spring maximum extension andk = 30 isthe spring constant.

The idea of our MD experiment is simple and is similarto experimentally designed reversibly switching surfaces[19]. First, we equilibrate a square (Lx= 30Lz = 20) ofa liquid film of thicknessLy 20 (they-axis is perpendic-ular to the film surface and periodic boundary conditionsare applied in thex, z-directions) consisting of 12000 par-ticles during teq = 5000 with the time integration steptMD = 0.01, which is used in the study. The tempera-ture 0.8 T 1.2 is controlled by means of a DPD ther-mostat with frictiondpd= 0.5 to preserve liquid motion.The film was positioned between two solid substrates con-sisting of three [0, 0, 1] fcc lattice layers of LJ atoms withthe shortest distance between the beads 22, 22 = 0.7,m2 = 10 and 22 = 0. The pressure in the system waskept close to the vapour pressure at given temperature Tby adjustingLy accordingly and making the second wallpotential at y = Ly purely repulsive. This has allowedfor a small gap between the wall and the liquid phase toestablish the gas phase. The solid wall particles were at-tached to anchor points via harmonic potential a= x

2,with the strength = 800 chosen such that the root-mean-square displacement of the wall atoms was smallenough to satisfy the Lindemann criterion for melting

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0 1 2 3 4 50.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

60 120 180

0

2

4

6

8

10

Surfacedensity,

[%]

Time

Static 12=0.65

t=0

t=2

t=4

t=10

t=20

Distribution

ofdensity

Distance from the substrate, y

FIG. 3: Evolution of the density distributions(y, t), the mainplot, and surface density (s(t) s(0))/s(0), the inset, atT = 0.8, NB = 5 after switching the interaction parameter12 from 0.2 to 0.65 at t0 = 100 during ts = 1, t= t t0.The solid (red) line in the inset is fit f1.

< r2 > = 0.1 22was shown to be sufficient to prevent the substrate fromhaving large and shear-rate divergent/dependent actualslip length[21]. The slip length measured in our experi-ments, as in[21], waslslip 24 11. After equilibration,parameter12 of the wall at y = 0 is changed from onevalue to another and we observe relaxation of interfacialparameters, including the surface tension.

The surface tension of a plane liquid-solid interface iscalculated according to [22], in the assumption of therigid solid substrate

LS= limym

ym0

Tt Tn y(y) d

dy

dy. (2)

Here(y) is distribution of density, (y) is the substratepotential generated by the solid wall particles, Tt,n(y) arethe tangential and normal components of the microscopicstress tensor evaluated according to [23], all quantitiesare averaged in the (x, z) plane. We note here that (2)is an approximation in our case of the weakly rough wallconsisting of moving particles, [22, 24, 25]. So that thenumerical procedure has been verified using the Young-Dupree equation by placing a substantially large cylindri-cal liquid drop (about 30000 particles) on the solid sub-strate and measuring the static contact angle0applyinga three-parameter circular fit (y y0)2 + (z z0)2 =R2to the free surface profile. The free-surface profile wasdefined in the study as the locus of equimolar points.The obtained values of0 were found to be within 3

ofthe contact angles calculated directly from the Young-Dupree equation using independently evaluated valuesof the surface tensions. The liquid-gasLV surface ten-sion has been calculated using large liquid drops (radius 30), similar to [26]. Typical dependencies of the in-tegrand of (2), LS(y), in static conditions are shown in

Fig. 1at different values of12.

In the dynamic experiments parameter 12 wasswitched from 0.2 to 0.65 (equivalently0 = 165

to 90

atT = 0.8,NB = 5) during ts= 1 with fixed12 = 0.7.The evolution of surface tension, density distribution and

surface density s = (Bym)1ym=50

(y)dy are shownin Figs. 2and3for different liquid compositions and tem-peratures. One can see that in general the relaxation is

very quick and almost independent of viscosity of the liq-uid at the first glance (the results are insensitive to lower-ingdpdto 0.3). Simple fit,f0 = C1+C2exp((tt0)/0),applied on average to normalised surface tension evolu-tion data reveals 0 = 7.8. The individual dependencies,inset Fig. 2, reveal more complex behaviour, which canbe approximated by f1 = C1 + C2exp((t t0)/1) +C3exp((t t0)/2)sin((t t0) + 0), TableI. One cansee that both1 and 2 are almost independent of molec-ular structure/viscosity despite the seventy-fold variationin . The observed values of 1 (the major relaxation)are close to the relaxation times found in the free surfacesof LJ liquids,[26], and thus correspond to the local relax-

ation on the length scale of the individual density peaks,Fig. 3,that is on the beads level rather than on the levelof the whole molecules. This is similar to the multi-scalerelaxation commonly observed in polymer dynamics,[27].In our case, initial, early times relaxation is defined bythe mean square displacement of individual monomersover relatively short distance of the order of the half ofthe distance between the density peaks (y = 0.5), Ta-ble I, while the liquid viscosity is defined by the muchslower molecular relaxation. This is also consistent withthe weak dependence on the destination value12(t > t0),TableI the last row.

The second, oscillatory relaxation,2and , is likely tobe due to the collective excitation of the particle motiontriggered by the sharp change of the solid wall potential,since the amplitude of oscillations decreases with increas-ing the switching time interval ts. In this case,2is sim-ply the time during which the excited wave of frequency travels some distance l2 comparable to the interfaciallayer width. Indeed, the product 2 = 2l2/2, where2 is the wave length, varies within 3.1 2 5.8,< 2 >= 4.4, TableI. Then on average < l2/2 > 0.7which means that the wave length of the excitations isroughly the width of the interfacial layer.

We would like to note that the observed weak depen-dence1,2() is in contrast to the relaxation time scalingLS

found in the MKT and in the interface forma-

tion theory. While the first peak density characteristictime scale found in the MD simulations[12]dp 16.5 isroughly comparable to our results, TableI, the observedweak dependence 1,2() rules out possible connectionsbetween LSand the MKT parameter (k

0)1.

The relaxation times revealed by the dynamic experi-ments directly imply that in the liquid compositions usedin our study, in the slow hydrodynamic motion, param-eter U LS/L > 1 is any macroscopic lengthscale) and surface tension is expected to be at equilib-

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T NB LV B 1 2

1

2

< r2M>

0.8 1 0.36 0.02 0.73 1.2 2.4 0.6 15.3 2.2 0.20 5.5 0.5 16.1 0.4 0.20 0.61

0.8 5 0.92 0.04 0.91 10.5 3.9 0.3 10.7 0.8 0.43 3.8 0.1 9.3 0.2 0.39 0.40

0.8 10 1.01 0.05 0.93 20.2 4.3 0.3 9.2 0.7 0.48 3.6 0.1 8.7 0.2 0.44 0.42

0.8 15 1.05 0.05 0.93 30.1 3.5 0.6 13.8 1.8 0.35 4.6 0.3 12.1 0.5 0.32 0.38

0.8 20 1.08 0.05 0.94 41.1 3.9 0.5 9.2 1.1 0.45 3.8 0.2 9.2 0.3 0.40 0.40

0.8 30 1.1 0.06 0.94 68.2 3.6 0.2 5.3 0.3 0.79 3.0 0.1 5.5 0.2 0.72 0.38

1.0 5 0.71 0.03 0.86 5.7 4.0 0.5 12.2 1.2 0.33 4.5 0.2 13.2 0.3 0.3 0.561.0 8 0.78 0.03 0.87 9.4 3.7 0.5 13.2 1.2 0.33 4.0 0.2 11.9 0.3 0.32 0.54

1.0 50 0.92 0.05 0.89 61.8 4.4 0.8 27.5 2.9 0.21 5.1 0.3 27.2 1.2 0.21 0.55

1.2 5 0.52 0.03 0.79 3.8 2.3 0.6 20.7 3.2 0.23 4.5 0.3 17 0.7 0.24 0.56

0.8 5 0.92 0.04 0.91 10.5 4.3 0.3 9.6 0.8 0.44 3.7 0.1 8.3 0.2 0.42 0.44

TABLE I: Parameters of the liquids (equilibrium surface tension LV, bulk densityB , dynamic viscosity) and characteristic

times of the liquid-solid interface formation (1,2, for the surface tension and 1,2,

for the surface density s applying fit

f1) at different molecular compositions (number of beads NB) and temperatures T. Viscosity was obtained as in [27]at the

bulk conditions. The last column is the end-monomer mean-square displacement

< r2M >during t= 1 across the interface

at the bulk conditions. 400 independent experiments. 12(t > t0) = 0.9.

FIG. 4: Snapshots and free surface profiles (the circular fits)in static and dynamic (U= 0.1) situations atT = 0.8,NB = 5and 12 = 0.9 (0 = 15

). The observed static and dynamiccontact angles are0= 123

and = 1384. The directionof the moving solid wall particles is indicated by the arrow.

rium. This in turn implies that the dynamic surface ten-sion is unlikely to be the cause of dynamic angle in ourcase, Sc

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some effect on the first layer of particles, the overall effectis not large, and both the surface tension and the densityare close to equilibrium, and far away from the values inthe case 12 = 0.37 (0 = 138

similar to the observeddynamic angle). How had then that dynamic angle (dif-ferent from 0 = 15

) been generated? We analysed thetangential force acting on the interface molecules in theregion (0 y ym= 2, zcl z zcl+ cl) at the con-tact line. We found that the tangential force f

clz acting onthe liquid from the solid substrate is concentrated within

cl and then drops significantly. This is not a coinci-dence, of course, since cl lslip. The value of the forceper unit length of the contact line is found to be sufficientto generate the observed contact angle according to thebalance of all forces acting on the contact line, the mod-ified Young-Dupree equation, that is fclz = 1.52 0.13

and from LVcos() = LS fclz , = 133 10. Butthis effect will need further studies.

In conclusion, we have directly established relaxationtime of the liquid-solid interfaces in a model system con-sisting of LJ molecules. The relaxation time, impor-tantly, appears to depend very weakly on the molecularstructure and viscosity and is found to be in such a rangethat interfacial tension LS should be in equilibrium in

slow hydrodynamic motion. This has been also verifiedin the MD experiments on dynamic wetting, where thedynamic contact angle was observed. Our results have di-rect repercussions on the theoretical interpretation andmodelling of the dynamic contact angle.Acknowledgement. This work is supported by the

EPSRC grant EP/H009558/1. The authors are gratefulto Y. Shikhmurzaev and T. Blake for useful discussions.

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