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Hydrodynamic Behavior at the Triple Line of Spreading Liquids and the Divergence Problem Alain Carre´* and Pierre Woehl Corning S.A., Fontainebleau Research Center, 7 bis, Avenue de Valvins, 77210 Avon, France Received November 26, 2001. In Final Form: January 24, 2002 A basic problem in spreading of a Newtonian liquid is the hydrodynamic description of the viscous energy dissipation near the moving contact line. With a sharp wedge profile, the braking force becomes infinite (divergence problem), implying that a drop is unable to spread on a solid surface, due to an infinite braking force. Introducing a “cutoff” length currently solves the problem, generating a finite value for the braking force. We propose an original approach to solve the divergence problem at a moving solid/liquid/ vapor triple line, exhibiting a fully explicit derivation for the viscous energy dissipation expression. It consists of observing that the rheological behavior of the liquid is modified near the triple line due to high shear rate. Above a critical value of the shear rate, near the triple line and near the solid surface, the liquid becomes shear-thinning, so that there is no divergence of the energy dissipation and viscous braking force. This description of the viscous braking phenomenon in liquid spreading is well supported by experiments of silicone oil spreading on glass substrates.
1. Introduction The wetting dynamics of liquids plays a key role in numerous applications, including coating, adhesive bonding, and printing processes. The equilibrium at the solid (S), liquid (L), and vapor (V) triple line is described by the well-known Young’s equation (1), in the form
γSV ) γSL + γ cos θ0
(1)
where γSV, γSL, and γ represent interfacial tensions respectively for the solid/vapor, solid/liquid, and liquid/ vapor interfaces and θ0 is the equilibrium contact angle at the solid/liquid/vapor (SLV) triple line. The contact angle as given by Young’s equation is a static equilibrium angle. However, during its motion toward an equilibrium shape, a liquid droplet scans a range of apparent (dynamic) contact angles. The dynamic angle depends on the rate of spreading, and several contact angle-velocity relationships have been proposed in the literature. Two basic models have been developed to describe the kinetics of spreading of liquids on rigid solid surfaces. The molecular kinetic theory proposed by Blake and Haynes2 considers a molecular “hopping” mechanism, based on Eyring rate process theory.3 The spreading of Newtonian liquids has been also largely described in the literature from the hydrodynamic theory.4-6 When a liquid spreads spontaneously on a flat and rigid solid, a dynamic equilibrium is set up in which the nonequilibrated Young force, and the corresponding excess capillary energy, causes triple line motion. Simultaneously, viscous dissipation in the liquid, near the wetting front, moderates the spreading speed. This theory has been successfully verified in wetting experiments in a regime of low spreading speeds (less than 10-3 m‚s-1), generated by (1) Young, T. Philos. Trans. R. Soc. 1805, A95, 65. (2) Blake, T. D.; Haynes, J. M. J. Colloid Interface Sci. 1969, 30, 421. (3) Glasstone, S.; Laidler, K. J.; Eyring, H. The Theory of Rate Processes; McGraw-Hill: New York, 1941. (4) de Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827. (5) Huh, C.; Scriven, L. E. J. Colloid Interface Sci. 1971, 35, 85. (6) Cox, R. G. J. Fluid Mech. 1986, 168, 169.
Figure 1. Description of a liquid wedge moving at a speed U on a solid substrate. The distance xm is a cutoff distance introduced to eliminate the divergence problem at the triple line.
capillary imbalance (expressed in eq 2) and relatively low contact angles. In these conditions, the hydrodynamic contribution to dissipation is considered to be dominant.7 Equilibrium at the triple line satisfies Young’s equation (eq 1). When the dynamic contact angle, θd, is greater than θ0, there is a net spreading force, Fm, per unit length of triple line. It is given by
Fm ) γSV - γSL - γ cos θd ) γ(cos θ0 - cos θd) (2) Equation 2 represents an unbalanced Young’s equation. As spreading occurs, the capillary energy, Ec, per unit of time and length of the SLV triple line, supplied is given by
Ec ) FmU
(3)
where U is wetting speed, defined as the ratio between total net liquid flow rate, Q (expressed per unit of length of the triple line), and wedge thickness, h (Q ) Uh, as shown in Figure 1). In the current hydrodynamic analysis of spreading of a Newtonian liquid, the boundary conditions in the lubrication approximation are assumed to be no slip at the solid surface, no shear at the free liquid surface, and the flow is assumed to be a noninertial (i.e., purely viscous) Stokes flow. This analysis in the wedge near the contact (7) Brochard-Wyart, F.; de Gennes, P. G. Adv. Colloid Interface Sci. 1992, 39, 1.
10.1021/la0117202 CCC: $22.00 © 2002 American Chemical Society Published on Web 03/27/2002
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line leads to a parabolic profile of speed, v(z), as a function of distance from the solid surface, z, given by
v(z) )
3U (2hz - z2) 2h2
(4)
where h is the height of liquid above the solid surface (Figure 1). The shear rate, γ˘ (z), can be simply obtained from eq 4 as
γ˘ (z) )
dv(z) 3U ) 2 (h - z) dz h
(5)
The viscous braking energy, Ed, per unit of time and length of the triple line, is obtained by integrating the product τ(z) γ˘ (z), where τ(z) is the shear stress (for a Newtonian liquid τ(z) ) ηγ˘ (z), where η stands for viscosity), leading to4
Ed )
3ηlU2 θd
(6)
where l is the logarithm of the ratio of the drop radius r divided by a microscopic cutoff length xm. It is supposed4 that, around xm, the liquid wedge is curved by long-range van der Waals forces (Figure 1). The cutoff length, xm, eliminates the divergence problem of the viscous braking force and energy dissipation in a sharp moving liquid wedge. Another method of removing this singularity is to allow for fluid slip in a small region near the moving contact line,5,8,9 with the boundary conditions at the front including a “slip length”. Another hypothesis to alleviate the force singularity is to assume that the substrate is never perfectly dry, but that evaporation and condensation at the front always produce a thin precursor layer.4,10,11 This hypothesis has recently found a renewed interest. Shanahan12,13 has developed a comprehensive approach that takes into account hydrodynamic and molecular mechanisms. It has been also demonstrated that the moving contact line paradox is alleviated for liquids exhibiting shearthinning behavior. At high shear stresses, fluids obey the Ellis constitutive law
( | | )
1 τ 1 ) 1+ η η0 τ1/2
R-1
where η is viscosity, η0 is the viscosity at zero shear stress, τ1/2 is the shear stress at which viscosity is reduced by a factor of 2, and R is a power law index.14 A hydrodynamic description near the contact line that includes a Marangoni mechanism leads to an acceptable prediction of the dynamic contact angle and removes the singularity (surface tension gradients along the gas-liquid interface generated by aging of the liquid near the triple line, contamination, and evaporation) as described by Shikhmurzaev.15,16 In this paper, we propose a more general alternative to solve the divergence problem at a moving triple line of a (8) Dussan, V. E. B.; Davis, S. H. J. Fluid Mech. 1974, 65, 71. (9) Hocking, L. M. J. Fluid Mech. 1977, 77, 209. (10) Hardy, W. B. Philos. Mag. 1919, 38, 49. (11) Bangham, D.; Saweris, S. Trans. Faraday Soc. 1938, 34, 561. (12) Shanahan, M. E. R. Langmuir 2001, 17, 3997. (13) Shanahan, M. E. R. Langmuir 2001, 17, 8229. (14) Weidner, D. E.; Schwartz, L. W. Phys. Fluids 1994, 6, 3535. (15) Shikhmurzaev, Y. D. Int. J. Multiphase Flow 1993, 19 (4), 589. (16) Shikhmurzaev, Y. D. AIChE J. 1996, 42 (3), 601.
Figure 2. Three regions in a liquid wedge moving at a speed U on a solid substrate. The factor n is the exponent of the power law describing the rheological behavior.
Newtonian liquid. The approach consists of realizing that the Newtonian behavior is lost above a critical shear rate. This behavior will be illustrated with the practical example of droplets of silicone oil spreading on glass substrates. The silicone oil is perfectly Newtonian below a critical shear rate, γ˘ c. Its viscosity decreases with increasing the shear rate above γ˘ c with a power law dependence. As a consequence, the divergence problem of energy dissipation is eliminated. The braking force and the energy dissipated in the liquid wedge are only described as a function of the rheological properties of the liquid. The transition from the Newtonian to the shear-thinning behaviors may indicate some type of structural order in the fluid under shear stress. When the system cannot respond fast enough to the deformation due to the shear stress, some degree of ordering begins to appear, resulting in a reduced viscosity. This behavior may be observed not only for polymeric molecules but also for short molecules such as alkanes.17,18 As an example, it has been shown from molecular simulations that viscosity of liquid alkanes has a shear-thinning behavior at high shear rates (above 108 s-1 for squalane (hexamethyltetracosane)), corresponding to extreme conditions of lubrication in an automotive engine.17 Such high shear rates are well above accessible experimental measurements. 2. Theoretical Section We consider the case where a Newtonian liquid becomes shear-thinning above a critical shear rate, γ˘ c. When the SLV triple line is moving on a solid surface at a speed U, the shear rate is at a maximum at the solid surface (z ) 0) and increases as x f 0. In this calculation, we will use the wedge profile given by the expression h ≈ xθd (Figure 2). The shear-thinning region is located close to the solid, resulting in a second-order influence on the LV interface profile. The distance from the SLV triple line, xST, below which the liquid is no longer Newtonian at the solid/liquid interface (z ) 0) may be defined from eq 5 and from γ˘ c as
γ˘ c(0) ) γ˘ c w xST )
3U γ˘ cθd
(7)
The liquid is Newtonian in region I of the liquid wedge (Figure 2, x > xST), where γ˘ (z) < γ˘ c. Region II, delimiting the Newtonian behavior for 0 < x < xST, may be defined from eq 5 as
(
γ˘ (z) e γ˘ c w z g λ ) xθd 1 -
) (
)
x x )h 1xST xST
(8)
(17) Moore, J. D.; Cui, S. T.; Cummings, P. T.; Cochran, H. D. AIChE J. 1997, 43, 3260. (18) Cui, S. T.; Cummings, P. T.; Cochran, H. D.; Moore, J. D.; Gupta, S. A. Int. J. Thermophys. 1998, 19, 449.
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Carre´ and Woehl
In region III, defined by 0 < x < xST and z < λ, the liquid is shear-thinning. Here, the shear stress, τ(z), may be simply modeled by a shear rate power law as
τ(z) ) aγ˘ (z)n
(9)
where a and n are constants and n < 1. For Newtonian behavior, n ) 1 and a becomes the viscosity. The speed profile in region II (Newtonian) is calculated by applying the speed profile described in the Introduction, namely, no slip at the solid and no shear at the LV interface. The interface between region II (Newtonian) and region III (shear thinning) then forms a discontinuity in the velocity and shear. A full description of this discontinuity is not possible within the framework of a full analytical solution. The speed profile in a liquid wedge of a shear-thinning liquid can be derived from the linear form of NavierStokes equation under the lubrication approximation, as being given by Carre´ et al.:19
1 2+ z n 1- 1v(z) ) U h 1 1+ n
( (
) [ ( )
1+1/n
) ]
xST
)(
)
1/nU
h
∫0 ∫0 τ(z) γ˘ (z) dz dx + x ∫0 ∫λh τ(z) γ˘ (z) dz dx + ∫xr ∫0h τ(z) γ˘ (z) dz dx ST
(12)
The first term of the sum corresponds to the shearthinning behavior (region III), the second term to region II (Newtonian behavior), and the third term to region I (Newtonian behavior). Integration of eq 12 with the respective expressions for the shear stress, τ(z) (eq 9), the shear rate, γ˘ (z) (eq 11), λ (eq 8), and xST (eq 7) leads to the following converging result:
31-na(2n + 1)n+1
U2 ηU2 3ηlU2 ‚ 1-n + + θd θd n (1 - n)(3n + 1 - n ) γ˘ c θd (13) n
2
with
l ) ln
( ) rθdγ˘ c 3U
(14)
Equations 13 and 14 define the energy dissipation only from the rheological properties (η,γ˘ c, a, n) and from the dynamic spreading parameters (U, θd, r). It is important to point out that there is no need for any ab initio parameter, such as the molecular cutoff length introduced in the classical hydrodynamic theory.4 The shear-thinning/Newtonian transition can also be obtained from the shear rate expression for a shearthinning behavior (eq 11) through defining xST and λ. This (19) Carre´, A.; Eustache, F. Langmuir 2000, 16, 2936.
λ ) xθd 1 -
x xST
n
(16)
The converging energy dissipation is then given by
Ed )
λ
ST
Ed )
(15)
γ˘ cθd
( ( ))
(11)
The energy dissipation may be estimated by considering the contributions from the three regions in the liquid wedge:
Ed )
1 2 + )U ( n )
and
1 z γ˘ (z) ) 2 + 1 n h
xST
leads to an alternative estimation of Ed. The calculation yields
(10)
The shear rate γ˘ (z) is
(
Figure 3. Rheological behavior of the PDMS oil as a function of the shear rate. Black squares represent the measured viscosity of the fluid. The solid line represents the linear fit rheological behavior with a sharp transition between Newtonian and shear-thinning behaviors.
a(2n + 1)2 ηU2 3ηlU2 U2 + ‚ 1-n + θd n(1 - n)(2 + n) γ˘ c θd nθd
with
l ) ln
(( ) ) rθdγ˘ c 1 2+ U n
(17)
(18)
Equations 15 and 17 have different analytical expressions. The difference between the two equations is due to the assumption of a sharp transition between Newtonian and shear-thinning behavior leading to a discontinuity in the velocity profile. However, the values of the energy dissipated are very close, because region I (third term on the right of eqs 13 and 17) is the main contributor to the dissipation and the factor l from eqs 14 and 18 does not differ substantially. As shown in the Experimental Section below, the domains defined by the xST and λ values (eqs 7, 8, 15 and 16) are very similar. It should be noted that the rheological behavior of fluids does not change sharply at a critical shear rate, but exhibits a transition region (Figure 3). Equations 13 and 17 allow a determination of the viscous braking force per unit of length of the triple line, Fd, satisfying Fd ) Ed/U. 3. Experimental Section The liquid used was a silicone oil (poly(dimethylsiloxane) (PDMS), DMS-T41, Gelest, Inc.). The liquid has a high molecular weight 62 700 g/mol, a surface tension of 21.5 mN/m, and a viscosity of 9.74 Pa‚s. The solid substrates were smooth, flat coverslips of sodalime glass (roughness (rms) below 0.3 nm). The substrates were cleaned by exposure to a flame or treated with
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Langmuir, Vol. 18, No. 9, 2002 3603
Figure 4. Comparison between the energy dissipation, Ed, and the capillary excess energy, Ec, for PDMS droplets spreading on glass. The dotted line represents the equality between Ed and Ec, and the open circles show experimental values.
Figure 5. Comparison between the energy dissipation, Ed, and the capillary excess energy, Ec, for PDMS droplets spreading on FDS-treated glass. The dotted line represents the equality between Ed and Ec, and the black circles show experimental values. perfluorodecyltrichlorosilane (FDS) to render the glass hydrophobic. The equilibrium wetting angle, θ0, with PDMS was close to zero degrees (°) after cleaning and 62° after FDS treatment. Contact angle measurements were obtained for the spreading of small droplets of 0.2 µL by using a contact angle goniometer equipped with a video camera in connection with a recorder and printer. Values of contact angles, θd, drop radius, r, and spreading speed, U, evaluated as dr/dt, were obtained as a function of time after droplet deposition, t. Measurements were taken at a temperature of 20 ( 2 °C and at 50% relative humidity. The rheological behavior of the PDMS oil is presented in Figure 3 (data provided by the supplier20). The liquid is considered to be Newtonian up to a shear rate of 1100 s-1. Above this critical shear rate, γ˘ c, the fluid viscosity is well modeled by a power law which may be written as
η ) 331.2γ˘ -0.5036
(19)
where the exponent represents n - 1 (n ) 0.4964). The analytical form of eq 19 may be derived from eq 9 as being
η)
τ(z) ) aγ˘ (z)n-1 γ˘ (z)
(20)
where η is the apparent viscosity of a shear-thinning fluid.
4. Results The excess of capillary energy was calculated during drop spreading from eq 3. The energy dissipation was (20) Arkles, B.; Vogelaar, G. In Metal Organics Including Silanes and Silicones. A Survey of Properties and Chemistry; Arkles, B., Ed.; Gelest Inc.: 1995; p 397.
Figure 6. Modeling of the liquid wedge, where the PDMS oil drop (r ) 0.593 mm) spreads on the glass at a speed of 40 µm/s and shows a dynamic contact angle of 0.96 rad. The curved solid line is obtained from eq 13 and the curved dotted line from eq 17.
calculated from eqs 13 and 14. The comparisons between the energy dissipation, Ed, and the excess capillary energy, Ec, are presented in Figure 4 for clean glass and in Figure 5 for the FDS-treated glass. Logarithmic scales are used, since the energy range covers several orders of magnitude. In both cases, although the contact angles are significantly different, very good agreement is observed for Ec and Ed. Figure 6 describes the liquid wedge with the different behaviors of the spreading liquid for a speed of 40 µm/s and a dynamic contact angle of 0.96 rad (clean glass). The transition at the solid/liquid interface from shear-thinning to Newtonian behavior occurs at a distance of 0.11 µm from the edge of the drop. The solid line defines the border between regions III and II, as obtained from eqs 7 and 8, and the dotted line from eqs 15 and 16. 5. Conclusions Several hypotheses exist that alleviate the divergence problem of the energy dissipation at a moving triple line. We propose a new approach for a fully explicit calculation of the energy dissipation, based on the fact that a Newtonian liquid may become shear-thinning near the triple line, above a critical shear rate. The distance xST defines the transition between the two rheological behaviors at the solid/liquid interface. This analysis differs from ref 14, where a shear-thinning behavior is considered over the entire range of shear rates. Our simple analysis considers a sharp transition between the Newtonian and the shear-thinning behaviors around a critical value of the shear rate, which leads to analytical approximate solutions. A more accurate treatment of the transition problem may be derived from matched asymptotic expansions or computed numeric solutions. The model was tested with spreading experiments of silicone oil on bare and silanized glass. For both substrates, the excess capillary energy is equal to the energy dissipated by viscous shearing. Therefore, the proposed analysis of viscous and dissipation phenomena in a liquid wedge moving on a solid substrate during drop spreading correlates with the experimental data, without requiring any adjustable parameter. As a general conclusion, this adaptation of the hydrodynamic theory is appropriate to describe the spreading of Newtonian liquid droplets on a rigid substrate. LA0117202
