PRL 110, 234503 (2013)
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PHYSICAL REVIEW LETTERS
Hydrodynamic Theory of Liquid Slippage on a Solid Substrate Near a Moving Contact Line E. Kirkinis* and S. H. Davis Department of Engineering Sciences and Applied Mathematics, Robert R. McCormick School of Engineering and Applied Science, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3125, USA (Received 8 January 2013; published 7 June 2013) In this Letter a hydrodynamic theory of liquid slippage on a solid substrate near a moving contact line is proposed. A family of spatially varying slip lengths in the Navier slip law recovers the results of past formulations for slip in continuum theories and molecular dynamics simulations and is consistent with well-established experimental observations of complete wetting. This formulation gives a general approach for continuum hydrodynamic theories. New fluid flow behaviors are also predicted yet to be seen in experiment. DOI: 10.1103/PhysRevLett.110.234503
PACS numbers: 47.55.np, 47.55.nd, 83.50.Lh, 83.50.Rp
Classical engineering and physical processes such as the spreading of adhesives, the coating of a solid surface by a liquid, the flow of lubricants, and oil recovery by water pumping in a deep well are characterized by a sharp interface separating a liquid from a gas or a second liquid phase. The intersection of this interface with a solid surface defines the contact line (CL) (cf. Fig. 1). The motion of this line has important consequences in modern technologies such as ink-jet printing in the context of polymers or cartilage growth [1], DNA screening [2], and the dynamics of the tear film at the cornea [3]. On the continuum scale the hydrodynamic motion of the CL with the standard no-slip condition of fluid mechanics gives rise to infinite local forces [4,5]. The microscopic physics that allows this motion to take place is still unknown, but molecular dynamics simulations [6] indicate that it is caused by slip, only in the vicinity of the moving contact line. Below we show how this key observation can be employed to develop a hydrodynamic theory of liquid slippage on a solid surface where slip is only active in the vicinity of the CL (cf. Fig. 1) and the no-slip condition is imposed away from it. The wedge domain of Fig. 1 represents the hydrodynamics of the local configuration of all complex moving CL problems and captures the local physics. Adopting the no-slip condition in a planar Stokes flow in a wedge, Huh and Scriven [4] showed that an infinite force is needed to move the contact line. In fact the local kinematics dictates that as one approaches the CL, the no-slip condition always leads to infinite force independent of the contact angle, inertia, or external forces because the stress components scale as 1=r as r ! 0, as was shown by Dussan V. and Davis [5], where r is the distance from the CL. Of course contact lines in nature do move without exerting infinite forces. It is thus necessary to introduce a unified model of slip valid over a range of materials and geometries that incorporates the aforementioned properties. In this Letter the flow singularity discussed in the previous paragraph is relieved by introducing a slip model in 0031-9007=13=110(23)=234503(5)
the context of the Navier slip law. In the context of our theory, this law is accompanied by a generalized Navier slip coefficient (or slip length) n ¼ n ðrÞ [7]. Consequently, we develop a detailed theory of complete spreading which results in slip velocities that encompass, as special cases, previous molecular dynamics results [6,8,9] and macroscopic slip models [10]. Quantitative agreement is obtained with well-established experimental results of complete spreading of silicon oils on a solid substrate [11] (cf. Fig. 2). To this end, we consider a circularly symmetric liquid droplet whose motion, in the vicinity of the CL, is described by the Navier slip law [7], vs ¼ n
1 @vr ; r @
(1)
FIG. 1. Upper right: coordinate system describing the geometry of a two-dimensional droplet in the vicinity of the contact line. (r, ) are plane polar coordinates denoting the location of a point in the liquid distance r from the CL which is located at ðx; yÞ ¼ ð0; 0Þ. The liquid-solid interface is located at ¼ 0. The liquid-vapor interface is located at ¼ . Lower left: in the frame of reference moving with the contact line the solid moves with velocity U to the right and the liquid-gas interface makes a dynamic contact angle ¼ with the solid substrate. The slip length n decreases to zero at the point r ¼ ‘=b1=n of the solidliquid interface and past this point it is replaced by the standard no-slip condition.
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FIG. 2 (color online). Log-log plot of capillary number versus dynamic contact angle (measured in degrees as opposed to radians in the main text) through Eq. (10) developed from our theory for the velocity exponent n ¼ 0:05 (continuous curve) and the experimental data of Hoffman [11] (circles and triangles) for two fluids that correspond to equilibrium contact angles eq ¼ 0. Most data are parallel to the dashed curve of slope 1=3. Inset: plot of the theoretical curve and experimental data on linear axes.
at the solid-liquid interface ( ¼ 0), where (r, ) are polar coordinates describing the planar liquid cross section as displayed in the upper right part of Fig. 1. vr is the radial velocity component and vs vr ð ¼ 0Þ. n is the slip coefficient (or slip length) which, in the context of the present formulation, is posed as n ¼
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PHYSICAL REVIEW LETTERS
PRL 110, 234503 (2013)
‘n bðn; Þr; rn1
(2)
where is the dynamic contact angle and bðn; Þ a scale factor to be determined by satisfying the boundary conditions. Within our model, n changes magnitude with distance r from the CL and it depends on a positive, real index n which also characterizes the flow stream function. Spatially dependent slip lengths, in the context of the Navier slip law, are not unknown in the thin liquid-film literature, for example in the context of viscous droplet spreading [12] or flow over patterned surfaces [13]. The angle characterizes the size of a moving wedge and relaxes to its equilibrium value, eq , when the wedge speed U vanishes. In this Letter we consider the case of complete spreading and so we assume that eq 0. The ! 0 asymptotics of the positive dimensionless scale factor bðn; Þ is such as to recover Stokes flow in an immobile wedge [14,15]. ‘ is a macroscopic scale that is determined by matching with an outer flow which satisfies the no-slip condition. The hydrodynamic model that follows from Eqs. (1) and (2) results in two distinct behaviors of stress and slip. First, an algebraically diverging shear stress rn1 ,
0 < n < 1 requires a finite corresponding slip length n that vanishes at the contact line in order to insure a finite slip velocity. The second case explores the opposite extreme where the slip length diverges close to the contact line (perfect slip) leading to a vanishingly small shear stress rn1 , n > 1 as the CL is approached. The case n ¼ 1 is special because both stress and slip are finite (but not zero) at the CL. In this Letter we consider n < 1. Further, the slip velocity we obtain with the present theory that scales as vs rn for general n can be considered as a generalization of other models or deductions met in the literature, for example those employed by Dussan V. [10] where the slip-velocity vs rp =ð1 þ rp Þ for integer or half-integer p. Other models introduced velocities of the form vs fðrÞ for a monotonically decreasing f [16]. The slip velocity of Ruckenstein and Dunn [17] vs r4 derived from the variation of the chemical potential due to coordinate dependence of the interaction potential at the liquid-solid interface. More recently, molecular dynamics simulations have derived slip velocities vs r1 [8] and vs Uð1 er ln2= Þ for a slip length [6,9]. Although the above behaviors seem incompatible with each other, in fact macroscopically they give rise to similar fluid behavior as was shown by Dussan V. [10]. Within the context of our theory the resulting slip velocities vs rn , n > 0 are finite and increase continuously until reaching the location r ¼
‘ b1=n
;
(3)
where n ¼ 0, at which point the no-slip condition must be imposed, causing the liquid to move with the solid velocity U at the liquid-solid interface. Thus the r < r flow can be asymptotically matched to its far-field counterpart, leading to uniformly valid spreading descriptions. Considering r as the characteristic scale of length, a characteristic velocity U 1 mm= sec, and Reynolds number Ur = 103 , where is the kinematic viscosity, we obtain for glycerine that r 0:68 mm and for water r 1 m. Consider a thin layer of a liquid that moves on a solid surface with velocity U. In the frame of reference fixed on the contact line, the solid moves with velocity U (cf. Fig. 1). Stokes flow in Fig. 1 is described by the biharmonic equation r4 c ¼ 0, for the stream function c ¼ c ðr; Þ where r, are cylindrical coordinates centered at the moving CL, ¼ 0 is the plane of the solid surface, and the radial and azimuthal velocities are related to the stream function by vr ¼ ð1=rÞð@ c =@Þ and v ¼ ð@ c =@rÞ, thus satisfying the incompressibility condition. Stream function solutions for a wedge (cf. Refs. [14,15]) have the general form c ¼ rnþ1 fðn; Þ. Thus the velocity and stress have the radial dependence vs rn and rn1 , respectively. For the motion of the CL the boundary condition of variable slip length in this frame is vr U ¼ n ð1=rÞð@vr =@Þ at the liquid-solid interface ¼ 0. The normal velocity v there must vanish.
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At the liquid-gas interface we require that the shear stress vanishes and the normal stress balances surface tension times curvature. By taking the interface to be planar at ¼ , we tacitly assume the asymptotic limit of small capillary number Ca ¼ U=, where is the dynamic viscosity and the liquid-gas surface tension. Application of the above boundary conditions leads to new wedge solutions with slip that satisfy a secular equation for the allowed contact angles with respect to the velocity exponent n, 4nbðn; Þ½cos2 ðÞ cos2 ðnÞ n sinð2Þ þ sinð2nÞ ¼ 0;
kinematic quantity n, and scale factor bðn; Þ that follows from Eqs. (4) and (5) for arbitrary dynamic angle . In Fig. 2 we display Ca versus by plotting the theoretical curve (10) compared with the experimental data of Hoffman [11] for completely spreading liquids, eq ¼ 0. In general there is an excellent agreement for Ca of the order of 0.1. There are departures from the scaling law when Ca becomes large, as expected. Asymptotically, Eq. (4) implies that bðn; Þ ¼ ð1=3Þ þ ð2=45Þðn2 þ 1Þ3 þ Oð4 Þ as ! 0. We thus obtain the asymptotic expression for the velocity U in Eq. (10),
(4) U¼
for n > 0, n 1 and sinð2Þ þ 2 cosð2Þ þ 4bð1; Þ sinð2Þ ¼ 0
(5)
for n ¼ 1. Even though in classical wedge studies the b ! 0 limit of Eqs. (4) and (5) provides a bifurcation criterion that relates admissible values with n (cf. Ref. [15]), in the present situation these equations determine the value of the scale factor b ¼ bðn; Þ that appears in expression (2) for the slip length. From the stream functions, it is easy to derive the slip velocity vs vr ð ¼ 0Þ of the liquid at the interface ¼ 0. Further, reverting to the lab frame of reference, vs acquires the form n r vs ¼ U 1 bðn; Þ (6) ‘ for 0 < r < r and n > 0. The force of traction pulling the liquid toward the dry region is FðÞ ¼ so sl cos;
(7)
whereas at equilibrium Fðeq Þ ¼ 0, where so and sl are the vapor-solid and liquid-solid surface tensions, respectively. Following Brochard and de Gennes [18], the friction force from the liquid onto the solid is also Z ‘=b1=n @vx F¼ dx: (8) @y 0
U : nb
n 3 ; 6
! 0;
(11)
which is identical to the one derived in the literature [18–20], where several sets of extra assumptions are made. The factor lnðL=aÞ [L, a being macroscopic and molecular length scales, respectively] that multiplies the capillary number in these works is here identified with the factor n1 . de Gennes reports that the value of this logarithm should be between 15 and 20, so the corresponding value of the exponent n should be between 1=20 and 1=15. This lower value, n ¼ 1=20, is exactly the one we employed in the theory developed here leading to Eq. (10) to match the experimental data of Hoffman in Fig. 2, although our result (10) is valid for arbitrary dynamic contact angles . Models alternative to the one developed in this Letter, such as the one by Cox [21], asymptotically reduce to a relation that involves a logarithmic ratio of scales emanating from a process of asymptotic matching and thus reproduces the results of de Gennes. Thus this logarithm is again related to our velocity exponent n. In passing we mention that since b and U 3 , in the limit ! 0, the ratio U=b scales as 2 and thus the force F in Eq. (9) vanishes. Now consider a spherical drop at dynamic contact angle using the same slip law (see Fig. 3). In this case Eq. (10) describes the evolution of the spreading radius R. Conservation of the volume of the drop gives ¼
For the flow under consideration we have that vx ¼ vr cos v sin. The substitution of the derived stream function into Eq. (8) gives F ¼
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PRL 110, 234503 (2013)
2R3 3 1 3 : cos þ cos 1 2 2 3sin3
(12)
(9)
Equating the expressions (7) and (9) leads to the main result discussed in this Letter, U ¼ ð1 cosÞbðn; Þn; (10) which is a general expression that describes the contactline velocity in terms of material parameters , ,
FIG. 3. A nearly flat spherical droplet of constant volume spreading on a solid. Here eq ¼ 0.
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FIG. 4 (color online). Log-log plot of capillary number versus dynamic contact angle (measured in degrees as opposed to radians in the main text) through Eq. (10) developed from our theory for various velocity exponents. This curve multiplicity could be used to model fluid behavior not yet met in experiment. Inset: plot of the same theoretical curves on linear axes.
Differentiating with respect to time t and setting dR=dt ¼ U enables one to eliminate R in favor of the dynamic contact angle ; we obtain a single equation describing the evolution of the latter during spreading of a drop that incorporates the local hydrodynamics through the parameter bðn; Þ, 1=3 d 2 2n ¼ dt 3
bðn; Þð1 cosÞ½1 32 cos þ 12 cos3 4=3 : cos½1 32 cos þ 12 cos3 12 sin4 (13)
Equation (13) can be simplified in the limit ! 0 in which we obtain that d=dt 13=3 . Solving, and defining the characteristic length scale L ¼ 1=3 , we obtain 3=10 L ; (14) Ut i.e., Tanner’s law [19,20,22]. From Eq. (14) we also obtain the time dependence of the drop radius as 1=10 Ut RL ; (15) L i.e., the same exponent as the one reported in experiments [22–24] and theory [22,25]. The slip law introduced in Eqs. (1) and (2) is intrinsically nonlocal. We expect the following to provide a good measure of the velocity: Z ‘=b1=n 1 @vr vr U ¼ (16) dr: r @ r
¼0
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The left-hand side is just rn fð0Þ U, while the right-hand side is rn1 ½f0 ð0Þ=nb½ð‘n =rn1 Þ br; i.e., it gives rise to the definition of the slip length, Eq. (2), multiplied by the strain rate ð1=rÞð@vr =@Þ ¼ rn1 f0 ð0Þ [apart from a multiplicative constant that can be absorbed by redefining the nonlocal law (16)]. This discussion is in agreement with the expectation, established by numerical simulations [6], that the breakdown of local hydrodynamics near the contact line should be alleviated by nonlocal slip. Thus far we discussed how the theory developed in this Letter recovers known results from the theoretical and experimental literature. However, it may also be used to predict features not yet seen in experiment. For example, varying the velocity exponent n one can generate Ca curves that may be used to model unconventional fluid behavior especially for higher Ca numbers; cf. Fig. 4. Finally, values of the velocity exponent n may be complex which would lead to the motion of a cascade of eddies (moving Moffatt vortices [14]) in the vicinity of the contact line. The authors are grateful to the anonymous referees for comments that improved the manuscript.
*[email protected] [1] B.-J. de Gans, P. C. Duineveld, and U. S. Schubert, Adv. Mater. 16, 203 (2004); T. Xu, K. W. Binder, M. Z. Albanna, D. Dice, W. Zhao, J. J. Yoo, and A. Atala, Biofabrication 5, 015001 (2013). [2] J. Jing et al., in Proc. Natl. Acad. Sci. U.S.A. 95, 8046 (1998); I. I. Smalyukh, O. V. Zribi, J. C. Butler, O. D. Lavrentovich, and G. C. L. Wong, Phys. Rev. Lett. 96, 177801 (2006). [3] K. L. Maki, R. J. Braun, P. Ucciferro, W. D. Henshaw, and P. E. King-Smith, J. Fluid Mech. 647, 361 (2010); K. Nong and D. M. Anderson, SIAM J. Appl. Math. 70, 2771 (2010). [4] C. Huh and L. E. Scriven. J. Colloid Interface Sci. 35, 85 (1971). [5] E. B. Dussan V. and S. H. Davis, J. Fluid Mech. 65, 71 (1974). [6] P. A. Thompson and M. O. Robbins. Phys. Rev. Lett. 63, 766 (1989). [7] E. Lauga, M. P. Brenner, and H. A. Stone, in Handbook of Experimental Fluid Dynamics, edited by C. Tropea, A. Yarin, and J. F. Foss (Springer, New York, 2007), Chap. 19. [8] T. Qian, X. P. Wang, and P. Sheng, Phys. Rev. Lett. 93, 94501 (2004). [9] M. Y. Zhou and P. Sheng, Phys. Rev. Lett.64, 882 (1990). [10] E. B. Dussan V., J. Fluid Mech. 77, 665 (1976). [11] R. L. Hoffman, J. Colloid Interface Sci. 50, 228 (1975). [12] H. P. Greenspan, J. Fluid Mech. 84, 125 (1978). [13] M. J. Miksis and S. H. Davis, J. Fluid Mech. 273, 125 (1994); L. Kondic and J. Diez, Colloids Surf. 214, 1 (2003); C. Y. Wang, Phys. Fluids 15, 1114 (2003). [14] H. K. Moffatt, J. Fluid Mech. 18, 1 (1964).
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[15] D. M. Anderson and S. H. Davis, J. Fluid Mech. 257, 1 (1993). [16] K. Jacobs, R. Seemann, G. Schatz, and S. Herminghaus, Langmuir 14, 4961 (1998). [17] E. Ruckenstein and C. S. Dunn, J. Colloid Interface Sci. 59, 135 (1977). [18] F. Brochard-Wyart and P. G. de Gennes, Adv. Colloid Interface Sci. 39, 1 (1992). [19] P. G. de Gennes, Rev. Mod. Phys. 57, 827 (1985).
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[20] P. G. de Gennes, F. Brochard-Wyart, and D. Que´re´, Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves (Springer, New York, 2004). [21] R. G. Cox, J. Fluid Mech. 168, 169 (1986). [22] L. H. Tanner, J. Phys. D 12, 1473 (1979). [23] A. M. Cazabat and M. A. C. Stuart, J. Phys. Chem. 90, 5845 (1986). [24] J. D. Chen, J. Colloid Interface Sci. 122, 60 (1988). [25] V. M. Starov, Colloid J. USSR 45, 1009 (1983).
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Hydrodynamic Theory of Liquid Slippage on a Solid Substrate Near a Moving Contact Line E. Kirkinis* and S. H. Davis Department of Engineering Sciences and Applied Mathematics, Robert R. McCormick School of Engineering and Applied Science, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3125, USA (Received 8 January 2013; published 7 June 2013) In this Letter a hydrodynamic theory of liquid slippage on a solid substrate near a moving contact line is proposed. A family of spatially varying slip lengths in the Navier slip law recovers the results of past formulations for slip in continuum theories and molecular dynamics simulations and is consistent with well-established experimental observations of complete wetting. This formulation gives a general approach for continuum hydrodynamic theories. New fluid flow behaviors are also predicted yet to be seen in experiment. DOI: 10.1103/PhysRevLett.110.234503
PACS numbers: 47.55.np, 47.55.nd, 83.50.Lh, 83.50.Rp
Classical engineering and physical processes such as the spreading of adhesives, the coating of a solid surface by a liquid, the flow of lubricants, and oil recovery by water pumping in a deep well are characterized by a sharp interface separating a liquid from a gas or a second liquid phase. The intersection of this interface with a solid surface defines the contact line (CL) (cf. Fig. 1). The motion of this line has important consequences in modern technologies such as ink-jet printing in the context of polymers or cartilage growth [1], DNA screening [2], and the dynamics of the tear film at the cornea [3]. On the continuum scale the hydrodynamic motion of the CL with the standard no-slip condition of fluid mechanics gives rise to infinite local forces [4,5]. The microscopic physics that allows this motion to take place is still unknown, but molecular dynamics simulations [6] indicate that it is caused by slip, only in the vicinity of the moving contact line. Below we show how this key observation can be employed to develop a hydrodynamic theory of liquid slippage on a solid surface where slip is only active in the vicinity of the CL (cf. Fig. 1) and the no-slip condition is imposed away from it. The wedge domain of Fig. 1 represents the hydrodynamics of the local configuration of all complex moving CL problems and captures the local physics. Adopting the no-slip condition in a planar Stokes flow in a wedge, Huh and Scriven [4] showed that an infinite force is needed to move the contact line. In fact the local kinematics dictates that as one approaches the CL, the no-slip condition always leads to infinite force independent of the contact angle, inertia, or external forces because the stress components scale as 1=r as r ! 0, as was shown by Dussan V. and Davis [5], where r is the distance from the CL. Of course contact lines in nature do move without exerting infinite forces. It is thus necessary to introduce a unified model of slip valid over a range of materials and geometries that incorporates the aforementioned properties. In this Letter the flow singularity discussed in the previous paragraph is relieved by introducing a slip model in 0031-9007=13=110(23)=234503(5)
the context of the Navier slip law. In the context of our theory, this law is accompanied by a generalized Navier slip coefficient (or slip length) n ¼ n ðrÞ [7]. Consequently, we develop a detailed theory of complete spreading which results in slip velocities that encompass, as special cases, previous molecular dynamics results [6,8,9] and macroscopic slip models [10]. Quantitative agreement is obtained with well-established experimental results of complete spreading of silicon oils on a solid substrate [11] (cf. Fig. 2). To this end, we consider a circularly symmetric liquid droplet whose motion, in the vicinity of the CL, is described by the Navier slip law [7], vs ¼ n
1 @vr ; r @
(1)
FIG. 1. Upper right: coordinate system describing the geometry of a two-dimensional droplet in the vicinity of the contact line. (r, ) are plane polar coordinates denoting the location of a point in the liquid distance r from the CL which is located at ðx; yÞ ¼ ð0; 0Þ. The liquid-solid interface is located at ¼ 0. The liquid-vapor interface is located at ¼ . Lower left: in the frame of reference moving with the contact line the solid moves with velocity U to the right and the liquid-gas interface makes a dynamic contact angle ¼ with the solid substrate. The slip length n decreases to zero at the point r ¼ ‘=b1=n of the solidliquid interface and past this point it is replaced by the standard no-slip condition.
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FIG. 2 (color online). Log-log plot of capillary number versus dynamic contact angle (measured in degrees as opposed to radians in the main text) through Eq. (10) developed from our theory for the velocity exponent n ¼ 0:05 (continuous curve) and the experimental data of Hoffman [11] (circles and triangles) for two fluids that correspond to equilibrium contact angles eq ¼ 0. Most data are parallel to the dashed curve of slope 1=3. Inset: plot of the theoretical curve and experimental data on linear axes.
at the solid-liquid interface ( ¼ 0), where (r, ) are polar coordinates describing the planar liquid cross section as displayed in the upper right part of Fig. 1. vr is the radial velocity component and vs vr ð ¼ 0Þ. n is the slip coefficient (or slip length) which, in the context of the present formulation, is posed as n ¼
week ending 7 JUNE 2013
PHYSICAL REVIEW LETTERS
PRL 110, 234503 (2013)
‘n bðn; Þr; rn1
(2)
where is the dynamic contact angle and bðn; Þ a scale factor to be determined by satisfying the boundary conditions. Within our model, n changes magnitude with distance r from the CL and it depends on a positive, real index n which also characterizes the flow stream function. Spatially dependent slip lengths, in the context of the Navier slip law, are not unknown in the thin liquid-film literature, for example in the context of viscous droplet spreading [12] or flow over patterned surfaces [13]. The angle characterizes the size of a moving wedge and relaxes to its equilibrium value, eq , when the wedge speed U vanishes. In this Letter we consider the case of complete spreading and so we assume that eq 0. The ! 0 asymptotics of the positive dimensionless scale factor bðn; Þ is such as to recover Stokes flow in an immobile wedge [14,15]. ‘ is a macroscopic scale that is determined by matching with an outer flow which satisfies the no-slip condition. The hydrodynamic model that follows from Eqs. (1) and (2) results in two distinct behaviors of stress and slip. First, an algebraically diverging shear stress rn1 ,
0 < n < 1 requires a finite corresponding slip length n that vanishes at the contact line in order to insure a finite slip velocity. The second case explores the opposite extreme where the slip length diverges close to the contact line (perfect slip) leading to a vanishingly small shear stress rn1 , n > 1 as the CL is approached. The case n ¼ 1 is special because both stress and slip are finite (but not zero) at the CL. In this Letter we consider n < 1. Further, the slip velocity we obtain with the present theory that scales as vs rn for general n can be considered as a generalization of other models or deductions met in the literature, for example those employed by Dussan V. [10] where the slip-velocity vs rp =ð1 þ rp Þ for integer or half-integer p. Other models introduced velocities of the form vs fðrÞ for a monotonically decreasing f [16]. The slip velocity of Ruckenstein and Dunn [17] vs r4 derived from the variation of the chemical potential due to coordinate dependence of the interaction potential at the liquid-solid interface. More recently, molecular dynamics simulations have derived slip velocities vs r1 [8] and vs Uð1 er ln2= Þ for a slip length [6,9]. Although the above behaviors seem incompatible with each other, in fact macroscopically they give rise to similar fluid behavior as was shown by Dussan V. [10]. Within the context of our theory the resulting slip velocities vs rn , n > 0 are finite and increase continuously until reaching the location r ¼
‘ b1=n
;
(3)
where n ¼ 0, at which point the no-slip condition must be imposed, causing the liquid to move with the solid velocity U at the liquid-solid interface. Thus the r < r flow can be asymptotically matched to its far-field counterpart, leading to uniformly valid spreading descriptions. Considering r as the characteristic scale of length, a characteristic velocity U 1 mm= sec, and Reynolds number Ur = 103 , where is the kinematic viscosity, we obtain for glycerine that r 0:68 mm and for water r 1 m. Consider a thin layer of a liquid that moves on a solid surface with velocity U. In the frame of reference fixed on the contact line, the solid moves with velocity U (cf. Fig. 1). Stokes flow in Fig. 1 is described by the biharmonic equation r4 c ¼ 0, for the stream function c ¼ c ðr; Þ where r, are cylindrical coordinates centered at the moving CL, ¼ 0 is the plane of the solid surface, and the radial and azimuthal velocities are related to the stream function by vr ¼ ð1=rÞð@ c =@Þ and v ¼ ð@ c =@rÞ, thus satisfying the incompressibility condition. Stream function solutions for a wedge (cf. Refs. [14,15]) have the general form c ¼ rnþ1 fðn; Þ. Thus the velocity and stress have the radial dependence vs rn and rn1 , respectively. For the motion of the CL the boundary condition of variable slip length in this frame is vr U ¼ n ð1=rÞð@vr =@Þ at the liquid-solid interface ¼ 0. The normal velocity v there must vanish.
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At the liquid-gas interface we require that the shear stress vanishes and the normal stress balances surface tension times curvature. By taking the interface to be planar at ¼ , we tacitly assume the asymptotic limit of small capillary number Ca ¼ U=, where is the dynamic viscosity and the liquid-gas surface tension. Application of the above boundary conditions leads to new wedge solutions with slip that satisfy a secular equation for the allowed contact angles with respect to the velocity exponent n, 4nbðn; Þ½cos2 ðÞ cos2 ðnÞ n sinð2Þ þ sinð2nÞ ¼ 0;
kinematic quantity n, and scale factor bðn; Þ that follows from Eqs. (4) and (5) for arbitrary dynamic angle . In Fig. 2 we display Ca versus by plotting the theoretical curve (10) compared with the experimental data of Hoffman [11] for completely spreading liquids, eq ¼ 0. In general there is an excellent agreement for Ca of the order of 0.1. There are departures from the scaling law when Ca becomes large, as expected. Asymptotically, Eq. (4) implies that bðn; Þ ¼ ð1=3Þ þ ð2=45Þðn2 þ 1Þ3 þ Oð4 Þ as ! 0. We thus obtain the asymptotic expression for the velocity U in Eq. (10),
(4) U¼
for n > 0, n 1 and sinð2Þ þ 2 cosð2Þ þ 4bð1; Þ sinð2Þ ¼ 0
(5)
for n ¼ 1. Even though in classical wedge studies the b ! 0 limit of Eqs. (4) and (5) provides a bifurcation criterion that relates admissible values with n (cf. Ref. [15]), in the present situation these equations determine the value of the scale factor b ¼ bðn; Þ that appears in expression (2) for the slip length. From the stream functions, it is easy to derive the slip velocity vs vr ð ¼ 0Þ of the liquid at the interface ¼ 0. Further, reverting to the lab frame of reference, vs acquires the form n r vs ¼ U 1 bðn; Þ (6) ‘ for 0 < r < r and n > 0. The force of traction pulling the liquid toward the dry region is FðÞ ¼ so sl cos;
(7)
whereas at equilibrium Fðeq Þ ¼ 0, where so and sl are the vapor-solid and liquid-solid surface tensions, respectively. Following Brochard and de Gennes [18], the friction force from the liquid onto the solid is also Z ‘=b1=n @vx F¼ dx: (8) @y 0
U : nb
n 3 ; 6
! 0;
(11)
which is identical to the one derived in the literature [18–20], where several sets of extra assumptions are made. The factor lnðL=aÞ [L, a being macroscopic and molecular length scales, respectively] that multiplies the capillary number in these works is here identified with the factor n1 . de Gennes reports that the value of this logarithm should be between 15 and 20, so the corresponding value of the exponent n should be between 1=20 and 1=15. This lower value, n ¼ 1=20, is exactly the one we employed in the theory developed here leading to Eq. (10) to match the experimental data of Hoffman in Fig. 2, although our result (10) is valid for arbitrary dynamic contact angles . Models alternative to the one developed in this Letter, such as the one by Cox [21], asymptotically reduce to a relation that involves a logarithmic ratio of scales emanating from a process of asymptotic matching and thus reproduces the results of de Gennes. Thus this logarithm is again related to our velocity exponent n. In passing we mention that since b and U 3 , in the limit ! 0, the ratio U=b scales as 2 and thus the force F in Eq. (9) vanishes. Now consider a spherical drop at dynamic contact angle using the same slip law (see Fig. 3). In this case Eq. (10) describes the evolution of the spreading radius R. Conservation of the volume of the drop gives ¼
For the flow under consideration we have that vx ¼ vr cos v sin. The substitution of the derived stream function into Eq. (8) gives F ¼
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2R3 3 1 3 : cos þ cos 1 2 2 3sin3
(12)
(9)
Equating the expressions (7) and (9) leads to the main result discussed in this Letter, U ¼ ð1 cosÞbðn; Þn; (10) which is a general expression that describes the contactline velocity in terms of material parameters , ,
FIG. 3. A nearly flat spherical droplet of constant volume spreading on a solid. Here eq ¼ 0.
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2
α
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0
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0.1
Theory, n=1e−3 Theory, n=1e−2 Theory, n=1e−1
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FIG. 4 (color online). Log-log plot of capillary number versus dynamic contact angle (measured in degrees as opposed to radians in the main text) through Eq. (10) developed from our theory for various velocity exponents. This curve multiplicity could be used to model fluid behavior not yet met in experiment. Inset: plot of the same theoretical curves on linear axes.
Differentiating with respect to time t and setting dR=dt ¼ U enables one to eliminate R in favor of the dynamic contact angle ; we obtain a single equation describing the evolution of the latter during spreading of a drop that incorporates the local hydrodynamics through the parameter bðn; Þ, 1=3 d 2 2n ¼ dt 3
bðn; Þð1 cosÞ½1 32 cos þ 12 cos3 4=3 : cos½1 32 cos þ 12 cos3 12 sin4 (13)
Equation (13) can be simplified in the limit ! 0 in which we obtain that d=dt 13=3 . Solving, and defining the characteristic length scale L ¼ 1=3 , we obtain 3=10 L ; (14) Ut i.e., Tanner’s law [19,20,22]. From Eq. (14) we also obtain the time dependence of the drop radius as 1=10 Ut RL ; (15) L i.e., the same exponent as the one reported in experiments [22–24] and theory [22,25]. The slip law introduced in Eqs. (1) and (2) is intrinsically nonlocal. We expect the following to provide a good measure of the velocity: Z ‘=b1=n 1 @vr vr U ¼ (16) dr: r @ r
¼0
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The left-hand side is just rn fð0Þ U, while the right-hand side is rn1 ½f0 ð0Þ=nb½ð‘n =rn1 Þ br; i.e., it gives rise to the definition of the slip length, Eq. (2), multiplied by the strain rate ð1=rÞð@vr =@Þ ¼ rn1 f0 ð0Þ [apart from a multiplicative constant that can be absorbed by redefining the nonlocal law (16)]. This discussion is in agreement with the expectation, established by numerical simulations [6], that the breakdown of local hydrodynamics near the contact line should be alleviated by nonlocal slip. Thus far we discussed how the theory developed in this Letter recovers known results from the theoretical and experimental literature. However, it may also be used to predict features not yet seen in experiment. For example, varying the velocity exponent n one can generate Ca curves that may be used to model unconventional fluid behavior especially for higher Ca numbers; cf. Fig. 4. Finally, values of the velocity exponent n may be complex which would lead to the motion of a cascade of eddies (moving Moffatt vortices [14]) in the vicinity of the contact line. The authors are grateful to the anonymous referees for comments that improved the manuscript.
*[email protected] [1] B.-J. de Gans, P. C. Duineveld, and U. S. Schubert, Adv. Mater. 16, 203 (2004); T. Xu, K. W. Binder, M. Z. Albanna, D. Dice, W. Zhao, J. J. Yoo, and A. Atala, Biofabrication 5, 015001 (2013). [2] J. Jing et al., in Proc. Natl. Acad. Sci. U.S.A. 95, 8046 (1998); I. I. Smalyukh, O. V. Zribi, J. C. Butler, O. D. Lavrentovich, and G. C. L. Wong, Phys. Rev. Lett. 96, 177801 (2006). [3] K. L. Maki, R. J. Braun, P. Ucciferro, W. D. Henshaw, and P. E. King-Smith, J. Fluid Mech. 647, 361 (2010); K. Nong and D. M. Anderson, SIAM J. Appl. Math. 70, 2771 (2010). [4] C. Huh and L. E. Scriven. J. Colloid Interface Sci. 35, 85 (1971). [5] E. B. Dussan V. and S. H. Davis, J. Fluid Mech. 65, 71 (1974). [6] P. A. Thompson and M. O. Robbins. Phys. Rev. Lett. 63, 766 (1989). [7] E. Lauga, M. P. Brenner, and H. A. Stone, in Handbook of Experimental Fluid Dynamics, edited by C. Tropea, A. Yarin, and J. F. Foss (Springer, New York, 2007), Chap. 19. [8] T. Qian, X. P. Wang, and P. Sheng, Phys. Rev. Lett. 93, 94501 (2004). [9] M. Y. Zhou and P. Sheng, Phys. Rev. Lett.64, 882 (1990). [10] E. B. Dussan V., J. Fluid Mech. 77, 665 (1976). [11] R. L. Hoffman, J. Colloid Interface Sci. 50, 228 (1975). [12] H. P. Greenspan, J. Fluid Mech. 84, 125 (1978). [13] M. J. Miksis and S. H. Davis, J. Fluid Mech. 273, 125 (1994); L. Kondic and J. Diez, Colloids Surf. 214, 1 (2003); C. Y. Wang, Phys. Fluids 15, 1114 (2003). [14] H. K. Moffatt, J. Fluid Mech. 18, 1 (1964).
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[15] D. M. Anderson and S. H. Davis, J. Fluid Mech. 257, 1 (1993). [16] K. Jacobs, R. Seemann, G. Schatz, and S. Herminghaus, Langmuir 14, 4961 (1998). [17] E. Ruckenstein and C. S. Dunn, J. Colloid Interface Sci. 59, 135 (1977). [18] F. Brochard-Wyart and P. G. de Gennes, Adv. Colloid Interface Sci. 39, 1 (1992). [19] P. G. de Gennes, Rev. Mod. Phys. 57, 827 (1985).
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[20] P. G. de Gennes, F. Brochard-Wyart, and D. Que´re´, Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves (Springer, New York, 2004). [21] R. G. Cox, J. Fluid Mech. 168, 169 (1986). [22] L. H. Tanner, J. Phys. D 12, 1473 (1979). [23] A. M. Cazabat and M. A. C. Stuart, J. Phys. Chem. 90, 5845 (1986). [24] J. D. Chen, J. Colloid Interface Sci. 122, 60 (1988). [25] V. M. Starov, Colloid J. USSR 45, 1009 (1983).
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