Hydrokinetic simulations of nanoscopic precursor films in rough channels
S. Chibbaro, L. Biferale, K. Binder, D. Dimitrov, F. Diotallevi, A. Milchev, S. Succi
aa r X i v : . [ phy s i c s . f l u - dyn ] J a n Hydrokinetic simulations of nanoscopic precursorfilms in rough channels
S. Chibbaro , L. Biferale , K. Binder , D. Dimitrov ,F.Diotallevi , A. Milchev , S. Succi Dept. of Mechanical Engineering, University of “TorVergata”, via del politecnico 1 00133, Rome, Italy Dipartimento di Fisica e INFN, Universita’ di Tor Vergata,Via della Ricerca Scientifica 1, 00133 Rome, Italy Institut fur Physik, Johannes Gutenberg Universitat Mainz,Staudinger Weg 7, 55099 Mainz Germany University of Food Technologies, 26 Maritza blvd., 4000Plovdiv, Bulgaria Istituto per le Applicazioni del Calcolo CNR V. Policlinico137, 00161 Roma, Italy
Abstract.
We report on simulations of capillary filling of high-wetting fluids innano-channels with and without obstacles. We use atomistic (molecular dynamics)and hydrokinetic (lattice-Boltzmann) approaches which point out clear evidence ofthe formation of thin precursor films, moving ahead of the main capillary front. Thedynamics of the precursor films is found to obey a square-root law as the main capillaryfront, z ( t ) ∝ t , although with a larger prefactor, which we find to take the samevalue for the different geometries (2D-3D) under inspection. The two methods showa quantitative agreement which indicates that the formation and propagation of thinprecursors can be handled at a mesoscopic/hydrokinetic level. This can be consideredas a validation of the Lattice-Boltzmann (LB) method and opens the possibility ofusing hydrokinetic methods to explore space-time scales and complex geometries ofdirect experimental relevance. Then, LB approach is used to study the fluid behaviourin a nano-channel when the precursor film encounters a square obstacle. A completeparametric analysis is performed which suggests that thin-film precursors may have animportant influence on the efficiency of nanochannel-coating strategies. ydrokinetic simulations of nanoscopic precursor films in rough channels
1. Introduction
Micro- and nano-hydrodynamic flows are prominent in many applications in materialscience, chemistry and biology [1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 10, 11]. A thoroughfundamental understanding as well as the development of corresponding efficientcomputational tools are demanded. The formation of thin precursor films in capillaryexperiments with highly wetting fluids (near-zero contact angle) has been reportedby a number of experiments and theoretical works [13, 14, 15, 16, 17], mostly inconnection with droplet spreading, and only very recently [18] for the case of capillaryfilling. In this latter case the presence of precursor films could help to reduce thedrag, and this could have an enormous economic impact as mechanical technology isminiaturized, microfluidic devices become more widely used, and biomedical analysismoves aggressively towards lab on a chip technologies. In this direction, patternedchannels [19] and more specifically ultra-hydrophobic surfaces have been considered [20,21]. At microscopic scales, inertia subsides and fluid motion is mainly governed by thecompetition between dissipation, surface-tension and external pressure. In this realm,the continuum assumption behind the macroscopic description of fluid flow goes oftenunder question, typical cases in point being slip-flow at solid walls and moving a contactline of liquid/gas interface on solid walls [1, 13]. In order to keep a continuum descriptionat nanoscopic scales and close to the boundaries, the hydrodynamic equations are usuallyenriched with generalised boundary conditions, designed in such a way as to collectthe complex physics of fluid-wall interactions into a few effective parameters, such asthe slip length and the contact angle [22, 23]. A more radical approach is to quitthe continuum level and turn directly to the atomistic description of fluid flows as acollection of moving molecules [24], typically interacting via a classical 6-12 Lennard-Jones potential. This approach is computationally demanding, thus preventing theattainment of space and time macroscopic scales of experimental interest. In between themacroscopic and microscopic vision, a mesoscopic approach has been lately developedin the form of minimal lattice versions of the Boltzmann kinetic equation [25, 26].This mesoscopic/hydro-kinetic approach offers a compromise between the two methods,i.e. physical realism combined with high computational efficiency. By definition,such a mesoscopic approach is best suited to situations where molecular details, whilesufficiently important to require substantial amendments of the continuum assumption,still possess a sufficient degree of universality to allow general continuum symmetriesto survive, a situation that we shall dub supra-molecular for simplicity. Lacking arigorous bottom-up derivation, the validity of the hydro-kinetic approach for supra-molecular physics must be assessed case-by-case, a program which is already countinga number of recent successes [27, 28, 29, 30]. The aim of this paper is two-fold. First,we validate the Lattice-Boltzmann (LB) hydro-kinetic method in another potentially’supramolecular’ situation, i.e. the formation and propagation of precursor films incapillary filling at nanoscopic scales. We employ both MD and hydrokinetic simulations,finding quantitative agreement for both bulk quantities and local density profiles, at all ydrokinetic simulations of nanoscopic precursor films in rough channels
2. Lattice-Boltzmann model
In this work we use the multicomponent LB model proposed by Shan and Chen [33].This model allows for distribution functions of an arbitrary number of components, withdifferent molecular mass: f ki ( x + c i ∆ t, t + ∆ t ) − f ki ( x , t ) = − ∆ tτ k h f ki ( x , t ) − f k ( eq ) i ( x , t ) i (1)where f ki ( x , t ) is the kinetic probability density function associated with a mesoscopicvelocity c i for the k th fluid, τ k is a mean collision time of the k th component (with ∆ t a time step), and f k ( eq ) i ( x , t ) the corresponding equilibrium function. The collision-timeis related to kinematic viscosity by the formula ν k = ( τ k − ). For a two-dimensional9-speed LB model (D2Q9) f k ( eq ) i ( x , t ) takes the following form [26]: f k ( eq )0 = α k n k − n k u eqk · u eqk (2) f k ( eq ) i = (1 − α k ) n k n k c i · u eqk (3)+ 12 n k ( c i · u eqk ) − n k u eqk · u eqk for i=1 . . . f k ( eq ) i = (1 − α k ) n k
20 + 112 n k c i · u eqk (4)+ 18 n k ( c i · u eqk ) − n k u eqk · u eqk for i=5 . . . c i ’s are discrete velocities, defined as follows c i = , i = 0 , (cid:16) cos ( i − π , sin ( i − π (cid:17) , i = 1 − √ (cid:16) cos [ ( i − π + π ] , sin [ ( i − π + π ] (cid:17) , i = 5 − α k is a free parameter related to the sound speed of the k th component, accordingto ( c ks ) = (1 − α k ); n k = P i f ki is the number density of the k th component. The massdensity is defined as ρ k = m k n k , and the fluid velocity of the k th fluid u k is definedthrough ρ k u k = m k P i c i f ki , where m k is the molecular mass of the k th component. Theequilibrium velocity u eqk is determined by the relation ρ k u eqk = ρ k U + τ k F k (6) ydrokinetic simulations of nanoscopic precursor films in rough channels U is the common velocity of the two components. To conserve momentum ateach collision in the absence of interaction (i.e. in the case of F k = 0) U has to satisfythe relation U = s X i ρ k u k τ k ! / s X i ρ k τ k ! . (7)The interaction force between particles is the sum of a bulk and a wall components.The bulk force is given by F k ( x ) = − Ψ k ( x ) X x ′ s X ¯ k =1 G k ¯ k Ψ ¯ k ( x ′ )( x ′ − x ) (8)where G k ¯ k is symmetric and Ψ k is a function of n k . In our model, the interaction-matrixis given by G k ¯ k = g k ¯ k , | x ′ − x | = 1 ,g k ¯ k / , | x ′ − x | = √ , , otherwise . (9)where g k ¯ k is the strength of the inter-particle potential between components k and ¯ k . Inthis study, the effective number density Ψ k ( n k ) is taken simply as Ψ k ( n k ) = n k . Otherchoices would lead to a different equation of state (see below).At the fluid/solid interface, the wall is regarded as a phase with constant numberdensity. The interaction force between the fluid and wall is described as F k ( x ) = − n k ( x ) X x ′ g kw ρ w ( x ′ )( x ′ − x ) (10)where ρ w is the number density of the wall and g kw is the interaction strength betweencomponent k and the wall. By adjusting g kw and ρ w , different wettabilities canbe obtained. This approach allows the definition of a static contact angle θ , byintroducing a suitable value for the wall density ρ w [34], which can span the range θ ∈ [0 o : 180 o ]. In that work [34], for the first time to the best of our knowledge,this phenomenological definition of the contact angle was put forward and the multi-component lattice Boltzmann method was used to study the displacement of a two-dimensional immiscible droplet subject to gravitational forces in a channel. In particular,the dynamic behavior of the droplet was analysed, and the effects of the contact angle,Bond number (the ratio of gravitational to surface forces), droplet size, and densityand viscosity ratios of the droplet to the displacing fluid were investigated. It is worthnoting that, with this method, it is not possible to know “a priori” the value of thecontact angle from the phenomenological parameters. Thus, an “a posteriori” map ofthe value of the static contact angle versus the value of the interaction strength g w hasto be obtained. To this aim, we have carried out several simulations of a static dropletattached to a wall for different values of g w [34, 35]. In particular, in our work, thevalue of the static contact angle has been computed directly as the slope of the contoursof near-wall density field, and independently through the Laplace’s law, ∆ P = γcosθH ,where H is the channel height. The value so obtained is computed within an error ∼ − ydrokinetic simulations of nanoscopic precursor films in rough channels k th component, the pressure is given by p k = ( c ks ) m k n k , where( c ks ) = (1 − α k ). To simulate a multiple component fluid with different densities, welet ( c ks ) m k = c , where c = 1 /
3. Then, the pressure of the whole fluid is given by p = c P k n k + P k, ¯ k g k, ¯ k Ψ k Ψ ¯ k , which represents a non-ideal gas law.The Chapman-Enskog expansion [26] shows that the fluid mixture follows theNavier-Stokes equations for a single fluid: ∂ t ρ + ∇ · ( ρ u ) = 0 , (11) ρ [ ∂ t u + ( u · ∇ ) u ] = − ∇ P + F + ∇ · ( µ ( ∇ u + u ∇ ))where ρ = P k ρ k is the total density of the fluid mixture, the whole fluid velocity u is defined by ρ u = P k ρ k u k + P k F k and the dynamic viscosity is given by µ = ρν = P k µ k = P k ( ρ k ν k ).To the purpose of analysing the physics of film precursors, it is important to noticethat , in the limit where the hard-core repulsion is negligible, both the Shan-Chenpseudo-potential and Van der Waals interactions predict a non-ideal equation of state,in which the leading correction to the ideal pressure is ∝ ρ . Therefore, both modelsobey the Maxwell area rule [35].
3. MD model
In the Molecular Dynamics simulation we use the simplest model, consisting of a fluid ofpoint-size particles that interact via a Lennard-Jones potential. Henceforth all lengthswill be quoted in units of σ , the atom diameter. The snapshot in Fig. 1b, illustrates oursimulation geometry. We consider a cylindrical nanotube of radius R = 11 and length L = 80, whereby the capillary walls are represented by densely packed atoms forminga triangular lattice with lattice constant 1 .
0. The wall atoms may fluctuate aroundtheir equilibrium positions, subjected to a finitely extensible non-linear elastic (FENE)potential, U F ENE = − ǫ w R ln (cid:16) − r /R (cid:17) , R = 1 . r is the distance between the particle and the virtual point which represents theequilibrium position of the particle in the wall structure, ǫ w = 1 . k B T , k B denotes theBoltzmann constant, and T is the temperature of the system. In addition, the wallatoms interact by a Lennard-Jones (LJ) potential, U LJ ( r ) = 4 ǫ ww h ( σ ww /r ) − ( σ ww /r ) i , (13)where ǫ ww = 1 . σ ww = 0 .
8. This choice of interactions guarantees no penetrationof liquid particles through the wall while in the same time the mobility of the wall atomscorresponds to the system temperature. The particles of the liquid interact with eachanother by a LJ-potential with ǫ ll = 1 .
40 so that the resulting fluid attains a density of ydrokinetic simulations of nanoscopic precursor films in rough channels ρ l ≈ .
77. The liquid film is in equilibrium with its vapor both in the tube as well as inthe partially empty right part of the reservoir. The interaction between fluid particlesand wall atoms is also described by a Lennard-Jones potential, Eq. (13), of range σ wl = 1and strength ǫ wl = 1 . δt = 0 . t where the MD timeunit (t. u.) t = ( σ m/ ǫ LJ ) / = 1 / √
48 and the mass of solvent particles m = 1.Temperature was held constant at T = 1 using a standard dissipative particle dynamics(DPD) thermostat [37, 38] with a friction constant ζ = 0 . r c = 1 . σ . All interactions are cut off at r cut = 2 . σ . Thetime needed to fill the capillary is of the order of several thousands MD time units.The top of the capillary is closed by a hypothetical impenetrable wall which preventsliquid atoms escaping from the tube. At its bottom the capillary is attached to arectangular 40 ×
40 reservoir for the liquid with periodic boundaries perpendicular tothe tube axis, see Fig. 1b. Although the liquid particles may move freely between thereservoir and the capillary tube, initially, with the capillary walls being taken distinctlylyophobic, these particles stay in the reservoir as a thick liquid film which sticks to thereservoir lyophilic right wall. At time t = 0, set to be the onset of capillary filling, weswitch the lyophobic wall-liquid interactions into lyophilic ones and the fluid enters thetube. Then we perform measurements of the structural and kinetic properties of theimbibition process at equal intervals of time. The total number of liquid particles is4 × while the number of particles forming the tube is 4800.
4. Numerical Results
We consider a capillary filling experiment, whereby a dense fluid, of density and dynamicviscosity ρ , µ , penetrates into a channel filled up by a lighter fluid, ρ , µ , see fig. 1. Forthis kind of fluid flow, the Lucas-Washburn law [39, 40] is expected to hold, at least atmacroscopic scales and in the limit µ ≫ µ . Recently, the same law has been observedeven in nanoscopic experiments [41]. In these limits, the LW equation governing theposition, z ( t ) of the macroscopic meniscus reads: z ( t ) − z (0) = γHcos ( θ ) Cµ t, (14)where γ is the surface tension between liquid and gas, θ is the static contact angle, µ is the liquid viscosity, H is the channel height and the factor C depends on the flowgeometry (in the present geometry C LB = 3 ; C MD = 2). The geometry we are going toinvestigate is depicted in fig. 1 for both models. It is important to underline that in theLB case, we simulate two immiscible fluids, without any phase transition.Since binary LB methods do not easily support high density ratios between thetwo species, we impose the correct ratio between the two dynamic viscosities through ydrokinetic simulations of nanoscopic precursor films in rough channels Figure 1.
Sketch of the geometry used for the description of the capillary imbibitionin the LB and MD simulations. (a) The 2 dimensional geometry, with length L + L and width H , is divided in two parts. The left part has top and bottom periodicboundary conditions, so as to support a perfectly flat gas-liquid interface, mimickingan “infinite reservoir”. In the right half, of length L , there is the actual capillary:the top and bottom boundary conditions are solid wall, with a given contact angle θ . Periodic boundary conditions are also imposed at the inlet and outlet sides. Themain LB parameters are: H ≡ ny = 40 , L = nz = 170 , ρ = 1; ρ = 0 . , µ =0 . , µ = 0 . , γ = 0 .
016 where H is the channel height, L is the channel length, ρ and ρ the gas and liquid densities respectively: µ k , k = 1 , γ the surface tension. (b) Snapshot of fluid imbibition for MD inthe capillary at time t = 1300 MD time-steps. The fluid is in equilibrium withits vapour. Fluid atoms are in blue. Vapour is yellow, tube walls are red and theprecursor is green. One distinguishes between vapour and precursor, subject to theradial distance of the respective atoms from the tube wall, if a certain particle has nocontact with the wall, it is deemed ’vapour’. The MD parameters are as follows [42]: R = 11 σ, L = 80 σ, ρ l = 0 . , µ = 6 . , γ = 0 . , σ = 1, where R is the capillaryradius and L its length. an appropriate choice of the kinematic viscosities. The chosen parameters correspondto an average capillary number Ca ≈ − and Ca ∼ . l cap and time in units of the capillary time t cap = l cap /V cap ,where V cap = γ/µ is the capillary speed and l cap = H/C LB for LB and l cap = R/C MD for MD. The reduced variables are denoted as ˆ z and ˆ t .In figure 2a, we show ρ ( z ) at various instants (in capillary units), for both MD andLB simulations. Choosing a constant time-interval ∆ˆ t = 7 between subsequent profiles,it is clear that the interface position advances slower than linearly with time. Therelatively high average density ρ ( z ) near the wall witnesses the presence of a precursorfilm attached to the wall. Indeed, the profiles ρ ( z ) at late times become distinctly ydrokinetic simulations of nanoscopic precursor films in rough channels t/t cap ( z ( t ) /l ca p ) LBMMDanalytical
Figure 2.
Dynamics of the bulk and precursor meniscus. (a) Position of the liquidmeniscus ˆ z ( t ) for LB and MD simulations. The position of the precursor film, ˆ z prec ( t )is also plotted for both models. ˆ z prec is defined as the rightmost location with density ρ = ρ bulk /
3. All quantities are given in natural “capillary” units (see text). Theasymptotic ( t > t cap ) rise of both precursor and bulk menisci follows a t / law, withdifferent prefactors (see the two straight lines), even though the underlying microscopicphysics is different. Notably, the precursor film is found to proceed with the law:ˆ z prec ( t ) = 1 . t . (b) Profiles of the average fluid density ρ ( z ) in the capillary atvarious times for LB and MD models. Figure 3.
The two figures show the fluid density profile in the vicinity of the meniscus,LJ-MD (left) and LB (right), at time ˆ t = 20 The MD results are rescaled so that thewidth is the same for both methods. nonzero far ahead of the interface position (near the right wall at ˆ z ≈
10 where thecapillary ends), due to a fluid monolayer attached to the wall of the capillary: thisprecursor advances faster then the fluid meniscus in the pore center, but also with a √ t law (see below). From this figure, it is appreciated that quantitative agreementbetween MD and LB is found also between the spatial profiles of the density field. Thisis plausible, since the LB simulations operate on similar principles as the MD ones,namely the fluid-wall interactions are not expressed in terms of boundary conditionson the contact angle, like in continuum methods, but rather in terms of fluid-solid(pseudo)-potentials. In particular, the degree of hydrophob/philicity of the two species ydrokinetic simulations of nanoscopic precursor films in rough channels √ t scaling in time, although with a larger prefactor than the front.As a result, the relative distance between the two keeps growing in time, with theprecursors serving as a sort of ’carpet’, hiding the chemical structure of the wall to theadvancing front. In spite of the different capillary number, the agreement between LBand MD can be attributed to the fact that these two approaches share the same static angle, θ = 0, since the “maximal film” [1], or complete wetting, configuration has beenimposed by increasing the strength of the wall-fluid attraction [42, 43], i.e. imposingthat the spreading coefficient S > z = ˆ z + ˆ t , (15)where we have inserted the value of cos ( θ ) = 1, corresponding to complete wetting.As to the bulk front position, fig.2a shows that both MD and LB results superposewith the law (15), while the precursor position develops a faster dynamics, fitted by therelation [18]: ˆ z prec = ˆ z + 1 . t . (16)Similar speed-up of the precursor has been reported also in different experimental andnumerical situations [15, 44]. The precursor is here defined through the density profile, ρ ( z ), averaged over the direction across the channel.In figure 3, we show a visual representation of the quantitative agreement betweenMD and LB dynamics: we present the density isocontour which can be imagined as theadvancing front. It is seen that the interface positions are in good agreement and alsodensity structures are very similar.These results achieve the validation of the LB method against the MD simulations.Moreover, as recently pointed out [42, 41, 18], our findings indicate that hydrodynamicspersists down to nanoscopic scales. The fact that the MD precursor dynamics quantitatively matches with mesoscopic simulations, suggests that the precursor physicsalso shows the same kind of nanoscopic persistence.Fig. 4 shows the whole interface in the vicinity of the meniscus and, in particular,the shape of the precursor computed with both methods. The plot emphasises thepresence of a precursor film at the wall. The agreement between the two methods isagain quantitative. The interface in both methods turns out to be about 5 units. Theprecursor film profile, defined by the isoline of points with density ρ = ρ bulk /
3, is splitin two regions. In the first that arrives until y ≈
7, it is fitted by a circular profile withcenter at z = 0 , y = 20 and radius r = 18. In the second, from y ≈ a Caz . In this formula, a is a characteristic molecularsize which is taken to be a = 1 in our case. This profile is obtained in the lubrication ydrokinetic simulations of nanoscopic precursor films in rough channels y LBMDCircular fittheoretical max. film
Figure 4.
The extrapolated interface position for both methods and the theoreticalprofile for “maximal film”. The x- and y- axis are normalised in order to reproducethe LB units. The colored region represents the whole interface. It is delimited, on theleft, by the solid line representing the isolines of the bulk density ( ρ = 0 .
9) and on theright by the one describing the beginning of the lighter fluid ( ρ = 0 . L p = 7 . − Ca − found in a recentnanoscale experiment [14]. P ( y ) ytheoretical fitLBE Figure 5.
Disjoining pressure in LB and its fit based upon analytical expressionare shown versus the distance from wall. The analytical expression for Van der Waalsforces is given by A πy , where is the Hamakar constant A = π kα l ( α s − α L ) and α s , α L are the polarizabilities of the solid and the liquid. In our case of total wetting, A > A = 0 . ydrokinetic simulations of nanoscopic precursor films in rough channels Figure 6.
A channel of length L = 200 LB units and width H = 41 LB units isstudied. The contact angles at the bottom and top walls are taken to be θ = θ = θ .An square obstacle is also present, whose dimensions are given by the height h and thewidth w . approximation considering Van der Waals interaction between fluid and walls [1], inthe case of “maximal film” (perfect wetting). Quantitative agreement is again foundbetween LB, MD and analytics, thus corroborating the idea of the Shan-Chen pseudo-potential as a quantitative proxy of attractive Van der Waals interactions in the lowdensity regime, where hard-core repulsion can be neglected. It is interesting to notethat the presence of the precursor film guarantees an apparent angle θ = 0 ◦ ,whereas theangle calculated from the circular fit of the bulk meniscus would be θ ≈ ◦ Since theseforces are, apparently, the key microscopic ingredient to be injected into an otherwisecontinuum framework, it is plausible to expect that LB should be capable of providing arealistic and quantitative description of precursor dynamics. The results of the presentsimulations confirm these expectations and turn them into quantitative evidence. Wehave also computed the disjoining pressure inside the film in order to corroborate theidea that LB is capable of correctly describing the dynamics of the precursor film. Suchpressure is related to the chemical potential and it is function of the distance from walls,that is of the thickness of the film [2]. In fig. 5, the disjoining pressure computed insidethe film in the LB is compared with the fit based upon the analytical expression given forVan-der-Waals forces. LB is a diffuse-interface model and, therefore, can not guaranteea pure hydrodynamical behaviour with ideal interfaces inside a thin film, such as theprecursor film experienced in this work. Nevertheless, is seems to assure a disjoiningpressure at least compatible with microscopic physics, with the correct divergence atthe wall. This appears to be consistent with the fact that real interfaces are found tobe diffuse also in experiments at macroscopic scales [15]. ydrokinetic simulations of nanoscopic precursor films in rough channels Figure 7.
Snapshots of density at different times t =20000 , , , , , In this section, we study the capillary filling in a nano-channel in presence of an obstacle.In particular, we want to analyse the effect of the precursor films on the dynamics ofthe fluid when crossing the obstacle. This test-case has been simulated through theLB model and the geometry is depicted in figure 6. First, we simulate a channel withhydrophilic walls which support a complete wetting ( θ = 0 ◦ ). An obstacle is put in themiddle of the channel length with dimensions w = h = 20 LB units. Six snapshots of thedensity evolution with time are represented in figure 7. Precursor films are visible aheadof the meniscus of the penetrating fluid at the bottom wall. Such film wets the obstaclebefore the front encounters it. The obstacle is large (its width is equal to the half ofthe channel) but the contact angle is small and therefore the front does not pin and isable to pass the obstacle in a relative short time, as known according to the Gibbs, orConcus-Finn, criterion [45, 46]. Nevertheless, the meniscus dynamics is deeply affectedby the presence of the obstacle. It is seen that the liquid on the top wall advances muchmore rapidly than the one on the bottom wall. In particular, at the beginning and atthe end of the obstacle the meniscus line is strongly distorted. After some time,We now perform a systematic series of simulations, where we vary the height of theobstacle in order to check to what extent the dynamics is affected by wall roughness.In figure 8, the detailed characteristics of the front dynamics are analysed. In figure 8a,it is possible to appreciate that, for complete wetting θ = 0 ◦ , the presence of a small ydrokinetic simulations of nanoscopic precursor films in rough channels z ( t ) ( L B un i t s ) time t (LB units)No Obsth=5h=10h=15h=20 (b) z ( t ) ( L B un i t s ) time t (LB units)h=0 θ =40h=10 θ =0h=10 θ =40h=10 θ =30h=10 θ =20h=10 θ =10 (c) z ( t ) ( L B un i t s ) time t (LB units)h=0 θ =40h=5 θ =0h=5 θ =40h=5 θ =30h=5 θ =20h=5 θ =10 Figure 8.
In all figures square obstacles (with width equal to the height) are consideredand the front position versus time is shown. (a) Different obstacle height are consideredfor θ = 0 ◦ . (b)The contact angle is changed (from 0 to 40 degrees) taking the heightof the obstacle constant and equal to h = 10. The front dynamics for the case with noobstacle and θ = 40 ◦ is also shown for comparison. (c) Same as in (b) but with h = 5. ydrokinetic simulations of nanoscopic precursor films in rough channels h = 5 = H/
8) is negligible. The front advances almost in the same way for thecases h = 0 ◦ and h = 5. This is possibly due to the presence in these cases of precursorfilms which wet the obstacle anticipating the meniscus. In this way, they “hide” theroughness and let the front pass through the obstacle without feeling any roughness. Foran obstacle of middle width h = 10, the front is strongly distorted and slowed down butit recovers the initial slope, after it has passed through the obstacle. On the contrary, forlarger obstacles h ≥
15 the dynamics of the meniscus seems to be irreversibly changedand it advances with a decreased slope after the obstacle. Then, we carry out numericalsimulations keeping the height of the obstacle ( h = 10) constant, while varying thecontact angle. In figure 8b, it is seen that the slope of the curve is strongly affected bythe value of the contact angle and, notably, for θ > ◦ the curve is not able to recoverthe initial dynamics but it results much slower. Moreover, the velocity of the advancingfront is strongly decreased when the walls are not very hydrophilic and with roughness.For instance, the front advances very similarly for θ = 40 ◦ without roughness h = 0 andfor h = 10 , θ = 10 ◦ . This seems to confirm the guess that the precursor films can reducethe drag in presence of roughness. In figure 8c, the same analysis is worked out for anobstacle with width w = 5. The effect of such obstacle is quite small for θ < ◦ , whereprecursor films are supposed to be present. Naturally, it results increasingly importantwith the increase of the contact angle and it causes a change in slope for θ > ◦ .
5. Conclusions
Summarising, we have thoroughly analysed the capillary filling in a nanochannel byusing atomistic and hydrokinetic methods.We have reported quantitative evidence of the formation and the dynamics ofprecursor films in capillary filling with highly wettable boundaries. The precursorshape shows persistent deviation from an ideal circular meniscus, due to the nanoscopicdistortion induced by the interactions with the walls. When properly scaled, theresults do not seem sensitive to geometry (in this work we investigate two differentgeometries) and resolution. This has been connected to the disjoining pressure inducedby Van der Waals interactions between fluid and solid and approximated in the LBapproach by a suitable phenomenological model. Our findings are supported by directcomparison between LB, MD simulations and theoretical predictions, which suggeststhat a continuum description (LB) together with a proper inclusion of the solid-fluidinteraction is able to reproduce this phenomenon. In this sense, this work provides acomplete assessment of the LB method for the study of nanochannel capillary filling.Then, a nanochannel with a square obstacle has been simulated via the LB model.The complete parametric analysis points out that for highly wetting walls ( θ < ◦ ) thepresence of precursor films make the presence of small obstacles (with a width smallerthan 1/8 of the channel height) negligible. Furthermore, results seems to indicate thatto reach high flow rate it is preferable to choose very hydrophilic and rough walls ratherthan to use very smooth but hydrophobic walls. ydrokinetic simulations of nanoscopic precursor films in rough channels
6. acknowledgments [1] P. G. de Gennes, Rev. Mod. Phys. , 827 (1985)[2] P. G. de Gennes, F. Brochard-Wyart, and D. Qu´er´e, Bubbles, Pearls, Waves (Springer, 2003).[3] B. Bhushan, J.N. Israelachvili, U. Landman, Nature , 607 (1995).[4] B. Zhao, J.S. Moore, and D.J. Beebe, Surface-directed liquid flow inside microchannels, Science , 1023 (2001).[5] G. Whitesides and A. D. Stroock, Flexible methods for microfluidics, Phys. Today 54, No. 6, 42(2001).[6] A. D. Stroock et. al., Science 295, 647 (2002).[7] G. Martic et al., A Molecular Dynamics Simulation of Capillary Imbibition Langmuir , 7971,(2002).[8] S. Supple and N. Quirke, Phys. Rev. Lett., , 214501, (2003).[9] L.-J. Yang, T.J. Yao, and Y.-C. Tai, The marching velocity of the capillary meniscus in amicrochannel, J. Micromech. Microeng., , 220, (2004).[10] W. Juang, Q. Lui, and Y. Li, Chem. Eng. Technol., , 716, (2006).[11] D. M. Karabacak, V. Yakhot, and K. L. Ekinci, Phys. Rev. Lett. , 254505 (2007).[12] P. Joseph et. al., Phys. Rev. Lett. , 156104 (2006).[13] D. Bonn, J. Eggers, J. Indekeu, J. Meunier, E. Rolley, Rev. Mod. Phys to appear.[14] H. Pirouz Kavehpour, Ben Ovryn, and Gareth H. McKinley, Phys. Rev. Lett. , 196104 (2003).[15] J. Bico, and D. Qu´er´e Europhys. Lett. , 348 (2003).[16] T.D. Blake, and J. De Coninck Advances in Colloid and Interface Science , 21 (2002).[17] F. Heslot, A.M. Cazabat, and P. Levinson Phys. Rev. Lett. , 1286 (1989).[18] S. Chibbaro, L. Biferale, K. Binder, D. Dimitrov, F. Diotallevi, A. Milchev, S. Succi, S. Girardo,D. Pisignano Europhys. Lett. to appear (2008).[19] H. Kusumaatmaja, C. M. Pooley, S. Girardo, D. Pisignano, and J. M. Yeomans, Phys Rev E
Phys Fluids
16, 12 (2004).[21] J. Hyvaluoma and J. Harting,
Phys. Rev. Lett , 360, (1997).[23] S. Reddy, P.R. Schunk, R.T. Bonnecaze,
Phys of Fluids
17, 122104 (2005).[24] M.P. Allen and D.J. Tildesley, ”Computer Simulation of Liquids”, Clarendon Press, Oxford(1987); D. Rapaport, ”The Art of Molecular Dynamic Simulations”, Cambridge University Press,Cambridge (1995).[25] R. Benzi, S. Succi, and Vergassola, Phys. Rep. , 145, (1992).[26] D.A. Wolf-Gladrow
Lattice-gas Cellular Automata and Lattice Boltzmann Models (Springer,Berlin, 2000).[27] J. Horbach, and S. Succi, Phys. Rev. Lett. , 224503 (2006).[28] M. Sbragaglia,R. Benzi, L. Biferale, S. Succi, and F. Toschi, Phys. Rev. Lett. (2006) 204503. ydrokinetic simulations of nanoscopic precursor films in rough channels [29] M.R Swift, W.R. Osborn, J.M. Yeomans, Phys. Rev. Lett. (1995) 830.[30] J. Harting, C. Kunert, H.J. Herrmann, Europhys. Lett. (2006) 328.[31] S. Cottin-Bizonne J.L. Barrat, L. Bocquet, E. Charlaix Nat. Mater. , 237-240 (2003).[32] N.R. Tas et al., Appl. Phys. Lett. , 3274, (2004).[33] X. Shan, and H. Chen, Phys Rev E , 1815, (1993). X. Shan, and G. Doolen, J. Stat. Phys. (9) 3203, (2002)[35] R. Benzi, L. Biferale, M. Sbragaglia, S. Succi, and F. Toschi, Phys. Rev. E (2006) 021509.[36] H. Huang, D. T. Thorne, M. G. Schaap, and M. C. Sukop, Phys. Rev. E , 066701 (2007).[37] T. Soddemann, B. D¨unweg, and K. Kremer, ”Dissipative particle dynamics: A useful thermostatfor equilibrium and nonequilibrium molecular dynamics simulations”, Phys. Rev. E, , 046702(2003).[38] P.J. Hoogerbrugge and J.M.V.A. Koelmann, ”Simulating microscopic hydrodynamic phenomenawith dissipative particle dynamics”, Europhys. Lett. , 155 (1992); P. Espanol, ”Hydrodynamicsfrom dissipative particle dynamics”, Phys. Rev. E , 1734 (1995).[39] E.W. Washburn, Phys. Rev. (1921) 273.[40] R. Lucas, Kolloid-Z (1918) 15.[41] P. Huber, K. Knorr, and A.V. Kityk, Mater. Res. Soc. Symp. Proc. 899E, N7.1 (2006).[42] D. I. Dimitrov, A. Milchev, and K. Binder, Phys. Rev. Lett. , 054501 (2007)[43] S. Chibbaro, The European Physical Journal E 27 01 (2008).[44] Popescu, S Dietrich, and G Oshanin, J. Phys.: Condens. Matter On the behavior of a capillary surface in a wedge , Appl. Math. Sci., ,292(1969); P. CONCUS, and R. FINN, On Capillary Free Surfaces in the Absence of Gravity , ActaMath. , 177(1974); R. FINN,