The influence of van der Waals forces on droplet morphological transitions and solvation forces in nanochannels
TThe influence of van der Waals forces on droplet morphological transitionsand solvation forces in nanochannels
F. Dutka and M. Napi´orkowski Institute of Physical Chemistry, Polish Academy of Sciences, ul. Kasprzaka 44/52, 01-224 Warszawa,Poland Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, ul. Ho˙za 69, 00-681 Warszawa,Poland (Dated: 18 November 2018)
The morphological phase transition between a sessile and lenticular shapes of a droplet placed in a nanochannelis observed upon increasing the droplet volume. The phase diagram for this system is discussed within themacro- and mesoscopic approaches. On the mesoscopic level, the van der Waals forces are taken into accountvia the effective interface potential acting between the channel walls and the droplet. We discuss the contactangle dependence on the droplet volume and the distance between the walls; this angle turns out to be smallerthan the macroscopic Young’s angle. The droplet presence induces the solvation force acting between thechannel walls. It can be both attractive and repulsive, depending on the width of the channel.PACS numbers: 47.60.Dx,68.03.Cd,68.05.-n,68.65.-k,83.50.HaKeywords: nanofluidics, phase transitions, van der Waals forces, line tension, solvation forces
I. INTRODUCTION
The progress in miniaturization of microfluidic systemsbrings new challenges for the theoretical description ofsuch systems. The behavior and manipulation of liq-uid droplets or gas bubbles (called the discrete phase)in a planar channel of micrometer size filled with immis-cible continuous phase is rather well understood . Inthe absence of electrostatic interactions and neglectingthe gravity (which plays minor role on these scale) thedroplet can be described by the macroscopic theory .When the size of the channel becomes smaller and itsheight is below 100 nm, the droplet shape cannot be de-scribed by the macroscopic theory; one has to take intoaccount the long-ranged van der Waals forces. They giverise to the effective interaction between the walls of thechannel and the droplet surface . In our mesoscopicdescription we consider droplets which do not touch thewalls of the channel, and a thin layer of continuous phaseseparating the walls and the droplet is present . Thistype of morphologies will be investigated in the followinganalysis. Fabrication of nanochannels and filling themwith liquid is already experimentally feasible ; carbonnanotubes are good examples of such nanocapillaries .The influence of the effective interface potential on theshape of the droplets in rectangular and circular capillar-ies has been usually investigated in two regions .One region corresponds to the droplet surface close to thewalls of the channel where the disjoining pressure domi-nates. The second region corresponds to droplet surfacelocated in the center of the capillary where the effect ofdisjoining pressure on the shape of the droplet can beignored.In our mesoscopic analysis we investigate the channelheights in the range 10-100 nm and determine the influ-ence of the effective interface potential on droplet shapefor any position of the droplet surface. We discuss in de- tail the geometry of the droplets, such as the thickness ofthe layer between the droplet and the walls of the chan-nel, as well as the change of the apparent contact angleas function of the increasing height of the channel.Many of the previous papers on the shapes of thedroplets in microchannels have focused on the dropletswhich were spread between the walls of the channel .In the present analysis we put stress on the morphologicaltransition between the sessile state (the droplet touchingonly one wall of the channel) and the lenticular state (thedroplet touching both walls of the channel). The phasediagrams displaying this transition are presented and dis-cussed, in Section II for the macroscopic approach and inSection III for the mesoscopic approach. In Section IVwe point at the role of the line tension when comparingthe macroscopic and mesoscopic approaches. We also dis-cuss the solvation force which emerges between thechannel walls, Section V. It turns out that both in macro-scopic and mesoscopic approaches the sign of this forcechanges upon increasing the channel width, turning fromrepulsive to attractive. We show that the solvation forceis zero in situation when the droplet can be inscribedin the circle whose center coincides with the symmetrypoint of the droplet. Section VI contains discussion. II. MACROSCOPIC DESCRIPTION
On the macroscopic level one can distinguish threegeneric equilibrium shapes of the A -fluid droplet placedin a flat channel filled with B -fluid, see Fig. 1. For sim-plicity, we consider a quasi-two dimensional system whichis translationally invariant in one direction. By the shapeof the droplet we mean the shape of its cross section per-pendicular to the direction in which the system is trans-lationally invariant. Three different morphological statesof the droplet can be characterized by the number of a r X i v : . [ c ond - m a t . s o f t ] J u l FIG. 1. Three generic macroscopic states of an A -fluid dropletin a planar channel filled with the B -fluid. The droplet cantouch zero (0), one (1) or two (2) walls, and the macroscopicstates are denoted accordingly. The distance between thewalls is 2 H . walls it remains in contact with: (0), the droplet doesn’ttouch any of the walls; (1), the droplet touches only onewall, and (2), the droplet touches both walls. In all threecases the shape of the droplet can be described by anarc of a circle. In case (0) the droplet forms a full circleand this state is called the circular state. In the case (1)the shape is a circular segment and we call it the sessilestate. The state (2) is called the lenticular (lens-shaped)state. Whenever the AB interface touches the wall itforms with it an apparent contact angle θ Y . We notethat generically by the contact angle one means the an-gle between the droplet interface and the wall, i.e. π − θ Y .Here we use the angle θ Y to stress that one of the typicalexperimental realizations of such a system is the channelfilled with liquid (the B -fluid) and its vapor representsthe A -fluid. So, the angle θ Y is the angle formed by thedroplet of liquid ( B ) deposited on a planar wall in ambi-ent conditions ( A ). We shall often refer to the Young’sequation cos θ Y = γ W A − γ W B γ , (1)where γ W A , γ W B , and γ are the wall- A fluid, wall- B fluid, and A -fluid - B -fluid surface tension coefficients,respectively. We consider the angles 0 (cid:54) θ Y (cid:54) π/ .We assume that the fluids A and B are immiscible andincompressible such that the bulk free energy of the sys-tem with a droplet relative to the energy of the channelcompletely filled by the phase B depends neither on theshape nor on the position of the droplet. It depends onlyon the cross-sectional area A to which we shall often referto as the droplet’s volume. To track the morphologicalphase transitions we analyze only the surface free ener-gies, which for the above three states are given by:Ω = 2 γ √ π √ A , Ω = 2 γ (cid:112) π − θ Y + sin θ Y cos θ Y √ A , Ω = 2 γH (cid:16) π − θ Y θ Y + sin θ Y + 12 cos θ Y AH (cid:17) . (2)We notice that the energy of the circular state is alwayslarger than the one corresponding to the sessile state, Ω > Ω . There are thus two competing equilibriumstates: sessile and lenticular. If however, the circularstates are imposed on the system, e.g., via the constrainton the droplet to be placed symmetrically with respectto the center plane of the channel, then one also allowsfor the circular – lenticular transition.For large enough volumes A the lenticular state haslowest surface energy, see Figs 2, 3. The equations FIG. 2. The phase diagrams illustrating the circular-lenticular ( a ), and the sessile-lenticular ( b ) first-order tran-sitions. The phase diagrams are plotted in the contact angle θ Y and the volume A variables. The solid lines denote thecoexistence curves and the dashed lines are the spinodals.The surface energies (Ω , Ω , Ω ) and volumes ( A max , A max , A min ) are given in Eq. (2) and Eq. (3), respectively. Ω = Ω and Ω = Ω determine the coexistence curves.The circular, sessile, and lenticular states cease to existfor areas A > A max , A > A max , A < A min , respec-tively, which are given by: A max = πH ,A max = H θ Y ) ( π − θ Y + sin θ Y cos θ Y ) ,A min = H (cid:16) π − θ Y cos θ Y − θ Y (cid:17) , (3)and determine the spinodal curves.The free energy profiles corresponding to the circular –lenticular and the sessile – lenticular transitions are plot-ted as function of A/H for the special choice of θ Y = π/ FIG. 3. Surface energy Ω as a function of an area A forthe circular-lenticular ( a ), and sessile-lenticular ( b ) transitionsfor θ Y = π/
4. The surface energies (Ω , Ω , Ω ) and areas( A max , A max , A min ) are given in Eq. (2) and Eq. (3), respec-tively. The symbols Ω and Ω (Eq. (4)) denote the ener-gies of a continuum of unstable states which connect smoothlyspinodal points. The angle α characterizing unstable stateschanges between 0 (cid:54) α (cid:54) θ Y . spinodal points are connected by the lines consisting ofparticularly constructed unstable states whose morpholo-gies interpolate smoothly between the stable states. Thefree energies and volumes of these states are denotedby Ω and A for the circular – lenticular transition,and Ω and A for the sessile – lenticular transition.The unstable morphologies can be characterized by onlyone parameter, the contact angle α which changes from0 (cid:54) α (cid:54) θ Y , see Fig. 3. For α = θ Y one has A = A min and A = A min (the lenticular state), while for α = 0 the volume A = A max for the circular – lenticular tran-sition, and A = A max for the sessile – lenticular transi-tion. The free energies of these particular unstable statesare given byΩ =2 γH π − α cos α , Ω =2 γH π − θ Y − α + cos θ Y (sin θ Y − sin α ))cos θ Y + cos α . (4)One could think about other choice of unstable statesbut these proposed here are characterized by only oneparameter and they smoothly interpolate between thespinodal points on Fig. 3. It turns out that such unstablestates appear also in the mesoscopic description. III. MESOSCOPIC DESCRIPTIONA. Equilibrium shape of the droplet
We assume the system to be translationally invariantin the y -direction and due to the invariance of the confin-ing walls in the x direction the equilibrium shape of thedroplet has to be symmetric with respect to axis parallelto the z -axis. We fix this symmetry axis at x = 0 andplace the walls of the channel at z = H , and z = − H (Fig. 4). FIG. 4. The schematic shape of the A -fluid droplet depositedin a planar channel of height 2 H and filled with B -fluid. Theshape of the droplet is described by two functions: z = f ( x )and z = − g ( x ) which connect smoothly at f ( x d ) = − g ( x d ) = d . We fix the symmetry axis of a droplet at x = 0. The shape of a droplet can be described by two functions z = f ( x ) and z = − g ( x ) which connect smoothly at z = d with f (cid:48) ( x d ) = g (cid:48) ( x d ) = −∞ . The parameter x d is definedimplicitly by equation d = f ( x d ) = − g ( x d ). The surfacefree energy of the droplet per unit length in y -directionequals H [ f, g ] = (cid:90) x d − x d d x (cid:110) γ (cid:112) f (cid:48) ( x )) + ω ( H − f ( x )) − ω ( H + f ( x ))+ γ (cid:112) g (cid:48) ( x )) + ω ( H − g ( x )) − ω ( H + g ( x )) (cid:111) , (5)where ω ( (cid:96) ) is the effective interface potential between aflat wall and the interface at a distance (cid:96) from it. Themodel of this potential stems from the microscopic den-sity functional analysis for the one component fluid inwhich the attractive parts of the fluid-fluid and wall-fluidinterparticle pair potentials are given by w ( r ) = − A F ( σ + r ) , w W ( r ) = − A W ( σ W + r ) , (6)where A F > A W > σ and σ W are related to the molecularsizes of the fluid and wall particles, e.g., for argon σ ≈ . nm . For this model the surface tension coefficientis given by γ = A F π σ ( ρ B − ρ A ) , (7)and the effective interface potential equals ω ( (cid:96) ) = ∆ ρ π (cid:104) ρ B A F σ ˆ ω ( (cid:96)/σ ) − ρ W A W σ W ˆ ω ( (cid:96)/σ W ) (cid:105) , (8)where ˆ ω ( (cid:96) ) = 1 − (cid:96) arctan 1 (cid:96) . (9)Here ∆ ρ = ρ B − ρ A , and ρ A , ρ B , ρ W are the A -fluid, B -fluid, and wall densities, respectively. After introducingdimensionless quantitiesˆ ρ = 12 (cid:16) − ρ A ρ B (cid:17) , ˆ A = ρ W A W ρ B A F , ˆ σ W = σ W σ (10)the effective interface potential reduces to ω ( (cid:96) ) = γ ˆ ρ (cid:104) ˆ ω (cid:16) (cid:96)σ (cid:17) − ˆ A ˆ σ W ˆ ω (cid:16) (cid:96)σ σ W (cid:17)(cid:105) . (11)The surface tension coefficient and the effective inter-face potential in Eqs (7) and (8) can be also obtainedfrom microscopic analysis of the two-component fluid ata planar wall for specific choice of parameters character-izing the interparticle interactions, see the Appendix.The macroscopic Young’s contact angle is given bycos θ Y = 1 + ω ( (cid:96) π ) γ , (12)where ω ( (cid:96) π ) is the only minimum of the effective interfacepotential and (cid:96) π is the thickness of the adsorbed layeron a planar substrate. The effective interface potential ω ( (cid:96) → → ∞ , and ω ( (cid:96) → ∞ ) →
0, so ω (cid:48) ( (cid:96) < (cid:96) π ) < ω (cid:48) ( (cid:96) > (cid:96) π ) >
0. As the result of the minimization of the Hamiltonianunder the constraint of the fixed volume A of the droplet A = (cid:90) x d − x d d x (cid:16) ¯ f ( x ) + ¯ g ( x ) (cid:17) (13)one obtains the following equations for the equilibriumshape of the droplet z = ¯ f ( x ) and z = − ¯ g ( x )¯ f (cid:48)(cid:48) ( x )(1 + ¯ f (cid:48) ( x ) ) / = − ¯ ω (cid:48) ( H − ¯ f ( x )) − ¯ ω (cid:48) ( H + ¯ f ( x )) − λ , ¯ g (cid:48)(cid:48) ( x )(1 + ¯ g (cid:48) ( x ) ) / = − ¯ ω (cid:48) ( H − ¯ g ( x )) − ¯ ω (cid:48) ( H + ¯ g ( x )) − λ , (14)where λ is the Lagrange multiplier, and ¯ ω ( (cid:96) ) = ω ( (cid:96) ) /γ .After one integration we obtain (we skip the bars)1 (cid:112) f (cid:48) ( x ) = − ω ( H − f ( x )) + ω ( H + f ( x )) + λf ( x ) + C , (cid:112) g (cid:48) ( x ) = − ω ( H − g ( x )) + ω ( H + g ( x )) + λg ( x ) + C . (15)The boundary conditions: f ( x d ) = − g ( x d ) = d ,f (cid:48) ( x = 0) = g (cid:48) ( x = 0) = 0 ,f (cid:48) ( x d ) = g (cid:48) ( x d ) = −∞ (16)give λ = f η ( f ) + g η ( g ) f + g ,C = − C = f g f + g (cid:16) η ( f ) − η ( g ) (cid:17) ,dη ( d ) − λ d + C , (17)where f = f ( x = 0), g = g ( x = 0), and η ( z ) = (1 + ω ( H − z ) − ω ( H + z )) /z . In the case of symmetric droplet,i.e. g = f , the parameter η ( f ) becomes equal to theLagrange multiplier.Inserting the parameters λ , C , C , and d as functionof f and g into Eq. (15)1 (cid:112) f (cid:48) ( x ) = 1 − f ( x ) η ( f ( x )) + λf ( x ) + C , (cid:112) g (cid:48) ( x ) = 1 − g ( x ) η ( g ( x )) + λg ( x ) + C . (18)one gets the equation (cid:90) f d d f | f (cid:48) ( x ) | = (cid:90) g − d d g | g (cid:48) ( x ) | = x d , (19)which renders the values of g = g ( f ). Thus the equilib-rium shape of the asymmetric droplet for a fixed volume A , Eq. (13), can be parametrized by one parameter, e.g. f which is a function of the volume f = f ( A ). B. Symmetric droplet
Typical droplet shapes encountered in droplet mi-crofluidics correspond to the lengths of the droplets whichare much larger then the channel height. Then thedroplet is symmetric with respect to the plane parallel tothe walls and located at the channel’s center . However,there are situations, e.g. in the flow-focusing method ofdroplet formation in which the droplet is symmetricallydeposited in channel and doesn’t touch the sidewalls .In this section we discuss the shapes of such symmetricdroplets. In particular, we investigate the dependence ofthe contact angle and the thickness of the films spannedbetween the walls and the droplet on the channel height.For given macroscopic contact angle θ Y , Eq. (1), andthe channel height 2 H , the volume A determines theshape of the droplet. In the symmetric case one has g = f and Eqs (13) and (18) give the following rela-tion A = f / (cid:90) d t u ( f , t ) (cid:112) ( η ( tf ) − η ( f ) , (20)where u ( f , t ) = √ t − f t (cid:16) η ( tf ) − η ( f ) (cid:17)(cid:114) − f t (cid:16) η ( tf ) − η ( f ) (cid:17) . (21)The function u ( f , t ) is finite for t ∈ [0 , η ( f ) is positive for f > f = f m denoted as η m = η ( f = f m ), see Fig. 5.For t → η ( tf ) − η ( f ) = − f η (cid:48) ( f )(1 − t )+ 12 f η (cid:48)(cid:48) ( f )(1 − t ) + . . . , (22)and it follows from Eq. (20) that A → ∞ for f → f m .Thus for given height of the channel, the minimal thick-ness of the film between the droplet and the wall (cid:96) m = H − f m is attained as A → ∞ . One can check that (cid:96) m < (cid:96) π , where (cid:96) π fulfills ω (cid:48) ( (cid:96) π ) = 0, and for increasing H the minimal film thickness behaves as (cid:96) m = (cid:96) π − cos θ Y ω (cid:48)(cid:48) π ( (cid:96) π ) 1 H + O (cid:16) H (cid:17) , (23)see Fig.6. FIG. 5. The Lagrange multiplier η ( f ) as a function of f for H = 100 σ . The inset shows the close-up of η ( f ) near itsminimum at f = f m . The surface tension coefficient and theeffective interface potential parameters are chosen such that θ = π/ (cid:96) π = 2 σ .FIG. 6. The minimal film thickness (cid:96) m (curve ( a )) and itsapproximation (curve ( b )), Eq. (23), as function of H for A →∞ . The inset shows the relative difference between the (cid:96) π and (cid:96) m . The surface tension coefficient and the effective interfacepotential parameters are such that θ Y = π/ (cid:96) π = 2 σ . Macroscopically, the apparent contact angle of thesymmetric droplet in the lenticular state is given bycos θ Y = HR , (24)where R is the radius of curvature of the droplet, and H is at the same time one half of the channel’ height and thehighest position of the droplet interface. In mesoscopicdescription, we define the contact angle in the same way,and as the radius of curvature we take the inverse of acurvature at z = 0cos θ = f (cid:16) η ( f ) + 2 ω (cid:48) ( H ) (cid:17) = 1 + ω ( H − f ) − ω ( H + f ) + 2 ω (cid:48) ( H ) f . (25)The effective interface potential ω ( z ) ∝ /z for z (cid:29) H/σ and A → ∞ the mesoscopic contactangle behaves ascos θ =1 + ω ( (cid:96) m ) − ω (2 H − (cid:96) m ) + 2 ω (cid:48) ( H )( H − (cid:96) m )= cos θ Y + cos θ Y ω (cid:48)(cid:48) π ( (cid:96) π ) 1 H − ω (2 H ) + 2 ω (cid:48) ( H ) H + O (cid:16) H (cid:17) , (26)see Fig. 7, FIG. 7. The cosine of mesoscopic contact angle θ (curve ( a ))and its approximation (curve ( b )), Eq. (26), as function of H for A → ∞ . The inset shows the relative difference betweenthe contact angle of the droplet and the Young’s angle. Thesurface tension coefficient and the effective interface potentialparameters are such that θ Y = π/ (cid:96) π = 2 σ . C. Morphological transition
The free energy profile Ω as function of volume A ob-tained in the mesoscopic analysis is shown on Fig. 8.The points marked with: tr , tr , sp sp
1, and sp A sp , Ω sp ), ( A sp , Ω sp ), and ( A sp , Ω sp ), respectively.The lines connecting the spinodal points correspond tounstable states. We notice that for A > A sp the dropletsof asymmetric shapes cannot exist in a flat channel.The droplet profiles fulfill Eq. (15) and can beparametrized by f , and g – the highest and lowest posi-tion of the interface; for symmetric droplets g = f . Forboth the circular – lenticular, and the sessile – lenticulartransitions the stable and metastable states are charac-terized by an increasing f and g as a function of A ,Fig. 9. FIG. 8. Surface energy Ω as function of volume A for sym-metric (green line) and asymmetric (red line) shapes for θ Y = π/
4. The points tr , tr , sp sp
1, and sp (cid:96) π = 2 σ , and H = 50 σ . We notice that as soon as the lenticular state is attainedthe parameter f remains practically constant; it doesn’tincrease more than 0 . σ , see also Fig. 6. Upon increasingthe volume of the droplets A there is a jump in f and g at the transition points, Fig. 10.In the circular state the distance between the droplet sur-face and the wall is large enough such that the effectiveinteraction between the wall and the droplet surface hasno effect on the shape of the droplet. In mesoscopic anal-ysis the situation in which the droplet surface is withinthe distance (cid:96) ≈ (cid:96) π to the wall corresponds to the droplet-wall contact in the macroscopic description. At spinodalpoint sp (cid:96) π ;also for the sessile state sp H − g (cid:29) (cid:96) π ,Fig. 10. For spinodal point sp FIG. 9. The highest f and the lowest g positions of thedroplet surface as function of the volume A for circular –lenticular, ( a ), and sessile – lenticular, ( b ), transitions. In ( b )we do not display the lenticular branch which is the same asin ( a ). The solid lines correspond to stable states, and dashedlines to the metastable and unstable states. The transitionpoints are marked with tr , tr , and the spinodal points with sp sp sp b ). The surface tension coefficient and theeffective interface potential parameters are such that θ Y = π/ (cid:96) π = 2 σ , and H = 50 σ . FIG. 10. The equilibrium shapes of the droplets in the circular– lenticular, ( a ), and the sessile – lenticular, ( b ), transitionsat A = A tr and A = A tr , respectively. Part ( c ) showsthe droplet shapes at spinodal points sp sp
1, and sp
2. Thesurface tension coefficient and the effective interface potentialparameters are such that θ Y = π/ (cid:96) π = 2 σ , and H =50 σ . IV. LINE TENSION
We have already noticed that the mesoscopic circularstates have the same energy as the macroscopic ones.On the other hand the mesoscopic sessile and lenticu-lar states have a lower free energy as compared to theirmacroscopic counterparts. The mesoscopic free energy,beside the contribution scaling with the surface of thedroplet, contains also a line contribution connected withtwo (in the sessile state) and four (in the lenticular state)three-phase contact lines extending in the y -direction .This line contribution was not taken into account inthe macroscopic description. For long-ranged van derWaals forces rendering continuous wetting transition andexploited in our analysis the line tension coefficient isnegative . The formula for the line tension coefficientcontains various contributions among which the mostsignificant one includes the interaction of the solid wallwith the interface detaching from the wall. According toEq. (4.4) in Ref. 30 it takes the form τ = 1tan θ Y (cid:90) ∞ (cid:96) π ω ( y ) , (27)and contributes roughly to one half of the value of theline tension coefficient. Although the authors in Ref. 30analyzed the behavior of line tension in the vicinity ofwetting temperature, where θ Y →
0, we use 2 τ as theestimate of the line contribution to the free energy stem-ming from a single three-phase contact line, also awayfrom the wetting point.The inclusion of the line tension into the macroscopicdescription results in the change of the value of the vol-ume A at which the morphological transitions take place.It can be calculated by solving equationsΩ ( A ) = Ω ( A ) + 8 τ Ω ( A ) + 4 τ = Ω ( A ) + 8 τ (28)for the circular – lenticular, and the sessile – lenticu-lar transitions, respectively. In the case of the circular– lenticular transition the values of A tr and Ω tr char-acterizing the morphological phase transition calculatedwithin the mesoscopic description are well approximatedby the values obtained within the macroscopic descrip-tion with the inclusion of the line tension contributions,Fig. 11.In the case of the sessile – lenticular transition this proce-dure leads to results presented on Fig. 12. The line ten-sion calculated in the full mesoscopic description turnsout to be smaller (more negative) than the approximatevalue 2 τ used within this simple approach.The parameter f characterizing the lenticular state,and therefore the contact angle θ (Eq. (25)), remain prac-tically independent of the area A . For the values of the FIG. 11. The volume ( I ) and free energy ( II ) of the circular– lenticular transition as function of H obtained within meso-scopic description, ( a ), and macroscopic description includingthe line tension contributions, ( b ). The values of the volumeand macroscopic free energy at morphological phase transi-tion without taking into account the line tension are markedwith superscripts ( M ). thermodynamic and geometric parameters considered inour analysis, and for H > σ ≈ nm , Fig. 7, theYoung’s contact angle is a very good approximation ofthe mesoscopic contact angle. The relative difference issmaller than one per million. The line tension coefficientmakes between 3% − .
5% of the total free energy for H within 50 σ − σ . One could thus expect that the macro-scopic description without including the line tension con-tributions would predict the values of the volume at thephase transition to be located within similar error mar-gin, i.e. below 3%. However, this is not the case and thedifference between the macroscopic and mesoscopic de-scription is more pronounced, between 5% − A ( M ) tr /A tr − A ( M ) tr ) and mesoscopic description ( A tr ) decreaseslike 1 /H , as expected. In case of the sessile – lenticulartransition the numerical results are less reliable due tonumerical errors induced by solving Eq. 19 and calculat-ing the droplet shape and its free energy in the sessilestate. FIG. 12. The volume ( I ) and free energy ( II ) of the sessile –lenticular transition as function of H obtained within meso-scopic description, ( a ), and macroscopic description includingthe line tension contributions, ( b ). The values of the volumeand the macroscopic free energy at morphological phase tran-sition including the line tension are marked with superscripts( M ).FIG. 13. The relative difference between the volume at thecircular – lenticular (red dots), and the sessile – lenticular(blue dots) transitions within macroscopic approach withouttaking into account the line tension, A ( M ) tr , and mesoscopicapproaches, A tr . V. SOLVATION FORCE
Insertion of the A -fluid droplet into the channel filledby the B -fluid changes the free energy of the system and, in particular, modifies the solvation force acting be-tween the sidewalls. The solvation force F is calculatedas F = − ∂ Ω /∂ (2 H ) at fixed volume of the droplet.In macroscopic description only the free energy of thelenticular state depends on the channel height, Eq. (2).The solvation force per unit length in the y -direction isthus non-zero and equals F ( M ) = − ∂ Ω ∂ (2 H ) = γ (cid:16)
12 cos θ Y AH − π − θ Y θ Y − sin θ Y (cid:17) = γR (cid:16) d − R sin θ Y (cid:17) = 2 d ∆ p − γ sin θ Y , (29)where 2 d is the length of the droplet-wall interface, R is the radius of curvature of AB interface, and∆ p = p A − p B = γ/R is the Laplace pressure, see Fig. 14. FIG. 14. Schematic shape of the droplet in macroscopic de-scription of the lenticular state. The length of the droplet-wall interface is denoted by 2 d , the radius of curvature of the A − B interface by R , and the contact angle by θ Y . Thesolvation force (per unit length in the y -direction) acting onthe upper wall contains contribution from the Laplace force F L = 2 d ∆ p = 2 dγ/R , where ∆ p is the Laplace pressure, andthe surface tension contribution denoted by F γ = γ . Accordingly, the solvation force is positive when the cen-ter of each arc of the circle forming the AB interface(points O and O in Fig. 14) and the corresponding in-terface are located on the same side of the droplet sym-metry axis perpendicular to the channel walls. Upon in-creasing the channel height the solvation force decreasesand becomes zero when the arcs centers O and O mergeon the symmetry axis, and becomes negative for largervalues of H , Fig. 15. We notice that - for larger values of H - the lenticular state becomes metastable against thesessile or circular state. In particular, for large enoughchannel height one has 2 d = 0, and the lenticular stateceases to exist.0In mesoscopic description, the free energy of the lenticular state, Eq. (5), is given by H [ ¯ f ] = 4 (cid:90) x d d x (cid:110) γ (cid:113) f (cid:48) ( x )) + ω ( H − ¯ f ( x )) − ω ( H + ¯ f ( x )) (cid:111) , (30)where z = ¯ f ( x ) describes the equilibrium shape of the droplet. Correspondingly, the solvation force is given by F = − ∂ H [ ¯ f ] ∂ (2 H ) = − ∂ H [ ¯ f ] ∂H = − (cid:90) x d d x (cid:110) ω (cid:48) ( H − ¯ f ( x )) − ω (cid:48) ( H + ¯ f ( x )) (cid:111) . (31)In agreement with the macroscopic analysis conclusionsthe solvation force is positive for small values of the chan-nel height and becomes negative for larger values, Fig. 16.For decreasing H the thickness of the film between thedroplet and the wall, (cid:96) = H − f decreases and thederivative ω (cid:48) ( (cid:96) ) becomes more negative; therefore thesolvation force can be positive. For higher values of H the thickness (cid:96) increases, ω (cid:48) ( (cid:96) ) becomes less negativeand the solvation force changes its sign. FIG. 15. The diagram in ( θ Y , H ) variables illustrating thechange of sign of the solvation force in the case of macroscop-ically analyzed lenticular state at fixed volume A . The redline denotes the sessile-lenticular transition above which thelenticular state is metastable. The shape of the droplet corresponding to F = 0 issuch that the radius of curvature R of the droplet at z = 0 equals x d , R = x d . In this situation, the A − B interface can be approximated by two arcs of the samecircle with the center at ( x = 0 , z = 0), Fig. 17. FIG. 16. The solvation force F as a function of H calculatedwithin mesoscopic (dots) and macroscopic (red line) analysis.The green line is introduced to guide the eye. The calculationwas done for A = 15000 σ ; the surface tension coefficient γ and the effective interface potential parameters are such that θ Y = π/ (cid:96) π = 2 σ .FIG. 17. The shape of the mesoscopic droplet correspondingto zero solvation force, F = 0. The shape of the droplet issuch that the radius of curvature of the interface R at z = 0equals x d , and the A − B interface can be approximated by twoarcs of the same circle with the center at ( x = 0 , z = 0). Thedroplet shape corresponding to F ( M ) = 0 in the macroscopicanalysis also exhibits this feature. VI. DISCUSSION
We have derived the phase diagrams for the circular-lenticular and the sessile-lenticular morphological tran-sitions of a droplet in a channel within two approaches:macroscopic and mesoscopic. Since the free energy ofthe sessile state is always smaller than that of the cir-cular state the former transition can be observed onlywhen droplet configurations which are symmetric withrespect to the center plane of a channel are imposed onthe system. e.g., via the appropriate constraint. Bothmorphological transitions are first-order and are accom-panied by the presence of metastable and unstable states.In mesoscopic description the free energy profile, Fig. 8,is qualitatively the same as in the macroscopic descrip-tion, Fig. 3. However, the macroscopic approach whichis not corrected by the inclusion of the line tension con-tributions, overestimates both the free energies and vol-umes at transition points up to 14% as compared to themesoscopic values, see Fig. 13. This comparison can besubstantially improved by including the contact angle de-pendent line tension coefficient, Figs. 11, 12.The long-ranged interparticle interactions taken intoaccount in analysis, Eq. (6), render the critical wettingtransition at a planar substrate and lead to negative linetension coefficient. The interparticle interactions lead-ing to the first-order wetting transition give positive linetension coefficient . We suppose that in this case thevalues of volumes characterizing the sessile –lenticulartransition will be larger than in the case of negative linetension coefficients. In addition to the the transitionpoints, also the spinodal points would change within themacroscopic description including the line tension con-tributions. Thus the analysis of the droplet states inthe nanochannels can give us a hint about the underly-ing interparticle interaction and the order of the wettingtransition.In the mesoscopic description there is always a layerof the host B -fluid separating the A -fluid droplet fromthe channel walls. This is the most profound differencebetween the mesoscopic description and its macroscopiccounterpart, where one allows for the droplet-wall inter-face. Nevertheless, also in the mesoscopic approach onecan define the contact angle θ , see Eq. (25). This angleapproaches the macroscopic Young’s angle θ Y for H → ∞ and droplet’s volume A → ∞ . For mesoscopic channelheights and large droplets ( A → ∞ ) this angle is smallerthan θ Y , Fig. 7. The difference θ Y − θ decreases with in-creasing height and its relative value is smaller than oneper mil already for H = 50 σ .In the mesoscopic description of the lenticular states oflarge droplets the film thickness between the droplet andthe wall (cid:96) = H − f is smaller than (cid:96) π , i.e., the thick-ness of the adsorption layer of the B -fluid on a planarsubstrate, Fig. 6. The difference (cid:96) π − (cid:96) decreases withincreasing channel height and for H > σ it is smallerthan 0 . σ . However, even this minor difference give riseto the positive (repulsive) solvation force, which is also present in macroscopic description. Approximating (cid:96) by (cid:96) π would incorrectly render the always negative (attrac-tive) solvation force, see Eq. (31).The predicted change of sign of the solvation force inthe lenticular state, also reported in Refs. 19 and 20,brings new issue in experimental micro- and nanoflu-idics. Suppose that one wall of the channel filled withthe B -fluid can move in the direction perpendicular toit. Inserting many identical droplets of the A -fluid offixed volume (with large enough distance between themto prevent their coalescence) will determine the distancebetween the walls of the channel. This height is a func-tion of number and the volume of the inserted droplets.Generally, the droplets of the A -fluid immersed in thechannel filled with the B -fluid can act as micro- or nan-odampeners (shock absorbers). Appendix: Effective interaction between a flat wall anddroplet surface
Consider an interface fluctuating near a planar wall,see Fig. 18. This interface separates the phases A and B rich in components 1 and 2, respectively. The thermody-namic state of the system corresponds to the coexistenceof these A and B phases of the binary mixture. Thesystem is invariant in y -direction and z = f ( x ) denotesthe position of the interface. The interfacial Hamiltonian FIG. 18. The two-component system at a planar substrate.The system is invariant in y -direction and z = f ( x ) is a fluid-fluid interface separating phases A , and B rich in component1 and 2, respectively. takes the form H AB [ f ] = (cid:90) ∞−∞ d x (cid:110) γ AB (cid:112) f (cid:48) ( x )) + ω AB ( f ( x )) (cid:17) (A.1)where γ AB is the surface tension coefficient, and ω AB ( (cid:96) )is the effective interface potential between the wall andthe interface located at the distance (cid:96) from it. We con-2sider the following model of long-ranged attractive inter-particle w ij ( r ) and wall-particle w iW ( r ) interactions w ij ( r ) = − A ij ( σ ij + r ) , w iW ( r ) = − A iW ( σ iW + r ) , (A.2)where i, j = 1 , A ij and A iW are positive; the positive param-eters σ ij and σ iW are related to the molecular sizes of thefluid and substrate particles. For this model the surfacetension coefficient is equal γ AB = π (cid:88) i,j =1 A ij σ ij ( ρ iB − ρ iA )( ρ jB − ρ jA ) , (A.3)and the effective interface potential ω AB ( (cid:96) ) = π (cid:88) i,j =1 ( ρ iB − ρ iA ) (cid:16) ρ jB A ij σ ij ˆ ω ( (cid:96)/σ ij ) − ρ W A iW σ iW ˆ ω ( (cid:96)/σ iW ) (cid:17) , (A.4)where ρ iA , ρ iB denote the number density of i th compo-nent in phases A and B , ρ W is the density of the wall,and ˆ ω ( (cid:96) ) = 1 − (cid:96) arctan 1 (cid:96) . (A.5)For the following choice of the amplitudes and molec-ular sizes A ij = (cid:112) A ii A jj , A iW = (cid:112) A ii A W W σ = σ ij , σ W = σ iW , i, j = 1 , γ AB = π σ (cid:16) (cid:88) i =1 (cid:112) A ii ( ρ iB − ρ iA ) (cid:17) ω AB ( (cid:96) ) = π (cid:16) (cid:88) i =1 (cid:112) A ii ( ρ iB − ρ iA ) (cid:17)(cid:16) (cid:88) j =1 ρ jB (cid:112) A jj σ ˆ ω ( (cid:96)/σ ) − ρ W √ A W W σ W ˆ ω ( (cid:96)/σ W ) (cid:17) . (A.7)Upon introducing the dimensionless quantitiesˆ ρ AB = 12 (cid:16) − (cid:80) i =1 √ A ii ρ iA (cid:80) i =1 √ A ii ρ iB (cid:17) , ˆ A AB = ρ W √ A W W (cid:80) i =1 √ A ii ρ iB , ˆ σ W = σ W σ (A.8) the effective interface potential reduces to ω AB ( (cid:96) ) = γ AB ˆ ρ AB (cid:104) ˆ ω AB (cid:16) (cid:96)σ (cid:17) − ˆ A AB ˆ σ W ˆ ω AB (cid:16) (cid:96)σ σ W (cid:17)(cid:105) , (A.9)which is exactly the form of the effective interface poten-tial for the one component system, see Eqs (11) and (61)in Ref. 28. ACKNOWLEDGMENTS
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