8042
Langmuir 2007, 23, 8042-8047
Spreading Dynamics and Dynamic Contact Angle of Non-Newtonian Fluids X. D. Wang,† D. J. Lee,‡,* X. F. Peng,§ and J. Y. Lai| Department of Thermal Engineering, School of Mechanical Engineering, UniVersity of Science & Technology Beijing, Beijing 100083, China, Department of Chemical Engineering, National Taiwan UniVersity, Taipei 106, Taiwan, Laboratory of Phase Change & Interfacial Transport Phenomena, Department of Thermal Engineering, Tsinghua UniVersity, Beijing 100084, China, and Department of Chemical Engineering, R&D Center for Membrane Technology, Chung Yuan Christian UniVersity, Chungli 32023, Taiwan ReceiVed January 15, 2007. In Final Form: May 12, 2007 The spreading dynamics of power-law fluids, both shear-thinning and shear-thickening fluids, that completely or partially wet solid substrate was investigated theoretically and experimentally. An evolution equation for liquid-film thickness was derived using a lubrication approximation, from which the dynamic contact angle versus the contact line moving velocity relationship was evaluated. In the capillary spreading regime, film thickness h is proportional to ξ3/(n+2) (ξ is the distance from the contact line), whereas in the gravitational regime, h is proportional to ξ1/(n+2), relating to the rheological power exponent n. The derived model fit the experimental data well for a shear-thinning fluid (0.2% w/w xanthan solution) or a shear-thickening fluid (7.5% w/w 10 nm silica in polypropylene glycol) on a completely wetted substrate. The derived model was extended using Hoffmann’s proposal for partially wetting fluids. Good agreement was also attained between model predictions and the shear-thinning fluid (1% w/w cmc solution) and shear-thickening fluid (10% w/w 15 nm silica) on partially wetted surfaces.
1. Introduction Wetting solid surfaces using liquids has broad applications in printing, painting, adhesion, lubrication, and spraying. Two types of theoretical models exist that address the wetting dynamics of Newtonian fluids (i.e., the molecular kinetics model1-4 and the hydrodynamic model5-13). Petrov et al.14 proposed a combined model that considers both molecular kinetics and hydrodynamics. Theoretical models all have difficulty dealing with mathematical singularity at the solid-liquid-gas contact line.15-19 Numerous polymer solutions and solid suspensions have nonNewtonian characteristics. A few theoretical studies examined * Corresponding author. E-mail:
[email protected]. Tel: +886-223625632. Fax: +886-2-23623040. † University of Science & Technology Beijing. ‡ National Taiwan University. § Tsinghua University. | Chung Yuan Christian University. (1) Blake, T. D.; Haynes, J. M. J. Colloid Interface Sci. 1969, 30, 421. (2) Blake, T. D. In Wettability; Berg, J. C., Eds.; Marcel Dekker: New York, 1993; pp 251-309 (3) Blake, T. D.; Coninck, J. D. AdV. Colloid Interface Sci. 2002, 96, 21-36. (4) Blake, T. D.; Shikhmurzaev, Y. D. J. Colloid Interface Sci. 2002, 253, 196-202. (5) Voinov, O. V. Fluid Dyn. 1976, 11, 714. (6) Cox, R. G. J. Fluid Mech. 1986, 168, 169-194. (7) Zhou, M. Y.; Sheng, P. Phys. ReV. Lett. 1990, 64, 882-885. (8) Ping, S, P.; Zhou, M. Y. Phys. ReV. A 1992, 45, 5694-5708. (9) Shanahan, M. E. R. Langmuir 2001, 17, 3997-4002. (10) Shanahan, M. E. R. Langmuir 2001, 17, 8229-8235. (11) Sharpe, M. R.; Peterson, I. R.; Tatum, J. P. Langmuir 2002, 18, 35493554. (12) Rame, E.; Garoff, S.; Willson, K. R. Phys. ReV. E 2004, 70, 031608. (13) Ranabothu, S. R.; Karnezis, C.; Dai, L. L. J. Colloid Interface Sci. 2005, 288, 213-221. (14) Petrov, J. G.; Radoev, B. P. Colloid Polym. Sci. 1981, 259, 753-760. (15) Huh, C.; Scriven, L. E. J. Colloid Interface Sci. 1971, 35, 85. Huh, C.; Scriven, L. E. J. Colloid Interface Sci. 1971, 35, 197. (16) Hoffman, R. L. J. Colloid Interface Sci. 1975, 50, 228-241. (17) Dussan, E. B. Ann. ReV. Fluid Mech. 1979, 11, 371-400. (18) de Gennes, P. G. ReV. Mod. Phys. 1985, 57, 827-863. (19) Kistler, S. F. In Wettability; Berg, J. C., Eds.; Marcel Dekker: New York, 1993; pp 311-429.
how non-Newtonian fluids spread over solid substrates. Carre´ and Eustache20 identified a relationship between the dynamic contact angle and contact line moving velocity for a drop of shear-thinning fluid. They concluded that spreading dynamics depended only on the power index, n, of a shear-thinning fluid. Neogi and Ybarra21 theoretically analyzed the spreading dynamics of Ellis fluids and Reiner-Rivlin fluids in a capillary spreading regime; however, they concluded that the rheological characteristics of non-Newtonian fluids did not impact their wetting dynamics. Starov et al.22 investigated theoretically the wetting dynamics of shear-thinning and shear-thickening fluids on completely wetted solid substrates. They arrived at an approximate solution that ignored the flow field near the contact line. Betelu and Fontelos23,24 numerically assessed the wave solution for completely wetting, shear-thinning fluids in the capillary spreading regime. Few experimental studies have explored the wetting dynamics of shear-thinning fluids. Carre´ and Eustache20 determined the spreading of two shear-thinning fluids, PDMS + silica and acrylic typographic ink, on glass slides. Rafai and Bonn25,26 indicated that the spreading exponents of xanthan solution (shear-thinning fluid) and polyacrylamide solution (normal stress fluid) on mica surface were lower than that predicted for Newtonian fluids (0.1, known as the Tanner law27), confirming the finding obtained by Starov et al.22 (20) Carre´, A.; Eustache, F. Langmuir 2000, 16, 2936-2941. (21) Neogi, P.; Ybarra, R. M. J. Chem. Phys. 2001, 115, 7811-7813. (22) Starov, V. M.; Tyatyushkin, A. N.; Velarde, M. G.; Zhdanov, S. A. J. Colloid Interface Sci. 2003, 257, 284-290. (23) Betelu, S. I.; Fontelos, M. A. Appl. Math. Lett. 2003, 16, 13151320. (24) Betelu, S. I.; Fontelos, M. A. Math. Comput. Model 2004, 40, 729734. (25) Rafai, S.; Bonn, D.; Boudaoud, A. J. Fluid Mech. 2004, 513, 77-85. (26) Rafai, S.; Bonn, D. Physica A 2005, 358, 58-67. (27) Tanner, L. H. J. Phys. D 1979, 12, 1473. Tanner, L. H. J. Phys. D 1979, 12, 1979.
10.1021/la0701125 CCC: $37.00 © 2007 American Chemical Society Published on Web 06/23/2007
Dynamics, Contact Angle of Non-Newtonian Fluids
Langmuir, Vol. 23, No. 15, 2007 8043
∂p ) -Fg ∂z
(4)
The boundary conditions comprise no shear at the liquid free surface and no slip at the solid substrate surface, and the force balance normal to the free surface follows the Young-Laplace equation:
u|z)0 ) 0 w|z)0 ) 0 Figure 1. Schematic of the advancing system with apparent contact angle (θ) and distance from the truncation point (xm).
∂u | )0 ∂z z)h
No comprehensive theoretical model exists that considers wetting dynamics for power-law, completely wetting, or partially wetting fluids. Experimental data are also largely lacking. This study investigates theoretically and experimentally the wetting dynamics of both shear-thinning fluids and shear-thickening fluids on completely and partially wetted substrates.
∂2h 1 1 ∂x2 p|z)h ) pg + σ + ) pg + σ ≈ R1 R2 ∂h 2 3/2 1+ ∂x ∂2h pg - σ 2 (5) ∂x
2. Theory 2.1. Thin Film Equation. The following shear stress versus shear rate dependence is considered in this work
τ ) kγ˘ n
∂p ∂ ∂u ) µ ∂x ∂z ∂z
( )
(3)
(28) Narhe, R.; Beysens, D.; Nikolayev, V. S. Langmuir 2004, 20, 12131221.
)
( ( ))
∂2h ∂x2
p ) Fg(h - z) + pg - σ
(6)
Substituting eq 6 into eq 3 yields
(| |
∂ ∂u n - 1∂u ∂3h ∂h k ) Fg - σ 3 ∂z ∂z ∂z ∂x ∂x
(2)
Equation 2 with n ) 1, 1 denotes a Newtonian fluid, a shear-thinning fluid, and a shear-thickening fluid, respectively. The fluid flow field near the contact line of a spreading liquid film with a small contact angle can be approximated using the equations of motion subjected to lubrication theory. That is, we make the following assumptions: (1) The liquid film is much thinner than its horizontal scale, such that the flow is 1D. (2) The viscosity of air is negligible. (3) The fluid is incompressible and completely wets the solid substrate. (4) The acceleration and inertial effects can be omitted because the local Reynolds number, Re ) (ujh/ν), is small in the thin film region. Variables uj, ν, and h are the cross-sectional average velocity in the thin film, the liquid kinematic viscosity, and the film thickness, respectively. Restated, in the present work, the moving velocity of a liquid drop is assumed to be slow enough for the velocity and pressure fields in the wetting drop to adjust themselves to steady-statelike distributions. A more comprehensive model based on a transient behavior description is available.28 Hence, the derived analytical solution can be used to approximate the dynamic wetting process. In the vicinity of a liquid front, the liquid flow field can be described using a Cartesian coordinate system in two dimensions (Figure 1). The horizontal velocity component of the fluid is u(x, z, t). Using a lubrication approximation, the flow equation can be stated as
(
)
In eq 5, pg is atmospheric pressure, σ is surface tension, and R1 and R2 are the two principal radii of the air-liquid interface. Integrating eq 4 with respect to z and applying the YoungLaplace condition yields
(1)
where k is a consistency coefficient and n is the power exponent. The apparent viscosity can be written as
µ ) kγ˘ n - 1
(
)
(7)
Integrating eq 7 with respect to z and using the no stress boundary condition at the free surface generates
(
)|
|
∂3h ∂h ∂3h ∂h ∂u 1 σ - Fg ) 1/nsign σ 3 - Fg ∂z k ∂x ∂x3 ∂x ∂x
(h - z)1/n
1/n
(8)
Integrating eq 8 with respect to z and using the no slip boundary condition yields the following velocity profile in the liquid film:
u)
(
)|
|
1 ∂3h ∂h ∂3h ∂h n sign σ Fg σ - Fg 1/n n + 1 k1/n ∂x ∂x3 ∂x ∂x3 {1}/{n}+1 (h - (h - z){1}/{n}+1) (9)
Therefore, the average velocity at point x is
uj )
(
3
)|
∫0h u dz ) 2n n+ 1 k11/nsign σ∂∂xh3 - Fg∂h ∂x
1 h
|
∂h ∂x
Fg
1/n
∂3h ∂x3
σ
h{1}/{n}+1 (10)
Substituting eq 10 into eq 9 yields
u)
z {1}/{n}+1 2n + 1 uj 1 - 1 n+1 h
( (
)
)
(11)
The mass balance of liquid in a segment of thickness dx over all z is
∂h ∂(huj) + )0 ∂t ∂x
(12)
8044 Langmuir, Vol. 23, No. 15, 2007
Wang et al.
Substituting eq 10 into eq 12 results in the following thin film equation:
) (| |
(
1 ∂3h ∂h ∂ ∂3h ∂h n sign σ Fg σ + ∂t 2n + 1 k1/n ∂x ∂x ∂x3 ∂x3 ∂h Fg 1/nh{1}/{n}+2 ) 0 (13) ∂x
)
This study discussed the limiting cases of eq 13, which ignored capillary and gravitational forces. That is, in the capillary spreading regime, σ . FgR2. In the gravitational spreading regime, σ , FgR2. 2.2. Capillary Spreading Regime. In the capillary spreading regime, eq 13 can be simplified to
( ) ( ) (| |
σ n ∂h + ∂t 2n + 1 k
∂3h ∂ ∂3h sign 3 ∂x ∂x ∂x3
1/n
1/n {1}/{n}+2
h
)
() ( ) (
σ n dh -U + dξ 2n + 1 k
sign
3
d h d {1}/{n}+2 d h h dξ3 dξ dξ3
1/n
h)
()
n σ 1 U 2n + 1 k
1/n
h{1}/{n}+2 sign
( )| | d3h d3h dξ3 dξ3
1/n
)
2n + 1 n k d3h d 3h hn+1 3 ) sign 3 U n σ dξ dξ
(16)
(17)
Taking h ) 0 at ξ ) 0, the solution of eq 17 for film thickness is
h)
h)
( ) ( ( ) ( k n U σ
k n U σ
1/{n+2}
(n + 2)3
) )
1/{n+2}
3(2n + 1)1 - n(1 - n)nn
1/{n+2}
(n + 2) 1-n
3(2n + 1)
3
ξ3/{n+2}
(n - 1)n
n < 1 (18a)
( ) (|
ξ3/{n+2} n > 1 (18b)
|
n Fg 1/n ∂h ∂ ∂h ∂h + sign - 1/nh{1}/{n}+2 ) 0 ∂t 2n + 1 k ∂x ∂x ∂x (19)
)
Introducing ξ ) x - Ut, eq 19 becomes
dh n Fg 1/n d dh 1/n {1}/{n}+2 -U + h ) 0 (20) dξ 2n + 1 k dξ dξ
( ) (( )
)
Integrating eq 20 over ξ yields
-
2n + 1 n k dh 1+n h ) U dξ n Fg
(
)
( )
k dh ) Un dξ σ
(21)
( )
1/{n+2}
)
ξ{1 - n}/{n+2}
n < 1 (23a)
)
ξ{1 - n}/{n+2}
n > 1 (23b)
3n+1(n + 2)1 - n
1/{n+2}
(2n + 1)1 - n(1 - n)nn
tgθ )
(
( )
k dh ) Un dξ σ
3
n+1
1/{n+2}
(n + 2)1 - n
1/{n+2}
(2n + 1)1 - n(n - 1)nn
When solving the flow field in a spreading film, a local “microscopic contact angle” (θw) at the contact line is required as the boundary condition. This angle, however, cannot be measured directly in experiments. Extrapolated contact angle θ (Figure 1) is commonly used in place of θw.29-31 In data analysis, truncated length xm (Figure 1) is required to acquire the dynamic contact angle based on experimental observations. Let the dynamic contact angle be equal to the inclination angle at xm. By adopting a small contact angle limit, eqs 23a and 23b yield
θ ≈ tgθ )
(
() dh dξ
) ξ)xm - Ut
3n+1(n + 2)1 - n
θ ≈ tgθ )
Equation 18 suggests that the film shape for power-law fluids depends on the power exponent n. Carre´ and Eustache20 noted in their tests of drop spreading that the shear-thinning PDMS + silica and ink fluids had a film shape of a half lemon. 2.3. Gravitational Spreading Regime. For the gravitational spreading regime, eq 13 can be simplified to
( )
)
(
() dh dξ
)
( ) k n U σ
1/{n+2}
(2n + 1)1 - n(1 - n)nn
1/{n+2}
n
(
(15)
or
( )(
tgθ )
)0
Integrating eq 15 over ξ results in the following expression:
2n + 1 n/{n+2} k 1/{n+2} 1/{n+2} ξ n Fg n * 1 (22)
The film thickness of power-law fluids depends on power exponent n in the gravitational spreading regime. 2.4. Dynamic Contact Angle. Differentiating film thickness (eq 18) with respect to ξ yields the inclination angle at ξ for the capillary regime as follows:
(14)
| |)
3
(
h ) [(n + 2)]1/{n+2} U
)0
Taking a frame moving with the contact line at U, eq 14 can be converted with a new variable ξ ) x - Ut: 1/n
Considering h ) 0 at ξ ) 0, the solution of eq 21 can be expressed as
) ξ)xm - Ut
)
3n+1(n + 2)1 - n
(2n + 1)1 - n(n - 1)nn
xm{1 - n}/{n+2}
( ) k n U σ
1/{n+2}
1/{n+2}
n < 1 (24a)
1/{n+2}
xm{1 - n}/{n+2}
n > 1 (24b)
3. Experimental Section 3.1. Experiment. Polypropylene glycol (PPG) solution (SigmaAldrich), the test Newtonian fluid, had a surface tension of 32.37 mN/m, viscosity of 0.1158 Pa s, and molecular weight (MW) of 725. The shear-thickening fluids tested in this study were 7.5% w/w 10 nm silica powders in PPG solution (Sigma-Aldrich) with a surface tension of 32.54 mN/m and 10% w/w 15 nm silica powders in PPG solution with a surface tension of 46.99 mN/m. The shear-thinning fluids tested were 1% w/w carboxymethylcellulose sodium salt (cmc, MW 700 000 Da, Sigma-Aldrich) with a surface tension of 39.02 mN/m and 0.2% w/w Xanthan solution with a surface tension of 48.41 mN/m. The surface tensions of all test liquids were measured using a Kru´ss Processor Tensiometer K12 (Kru´ss GmbH, Hamburg, Germany). Clean microscope glass slides and mica were the test solid substrates. All experiments were conducted using the same slide; the glass slide was cleaned completely using detergent, ultrapure water, 99.5% ethanol, 2 N nitric acid, and acetone with ultrasonication.
Dynamics, Contact Angle of Non-Newtonian Fluids
Langmuir, Vol. 23, No. 15, 2007 8045
Figure 2. Apparent viscosity versus shear rate plots for the four tested liquids.
Figure 3. Repeated four wetting tests of drop radii versus time data. PPG + 7.5% w/w 10 nm silica solution. Substrate: glass slide. The mica foils were split carefully to obtain clean, smooth surfaces. Before each experiment, dry nitrogen was used to dry the slide or mica. A cone-plate rheometer (Advanced Pheometric Expansion System) measured the apparent viscosity versus shear rate relationship for all test liquids (Figure 2). The PPG + silica solutions have shearthickening behaviors, whereas cmc and xanthan solutions have shearthinning behaviors. The consistency coefficient and power exponent in eq 2 were obtained using regression rheological data for each fluid (Table 1). The static contact angle and drop-spreading dynamic tests were conducted using an FTA125 dynamic contact angle analyzer (Contact Angle and Surface Tension Instruments, Portsmouth, NH). A drop (4 µL) of selected liquid was deposited from the top of the horizontally
Figure 4. (a) R(t) and (b) θD(U) measured for the PPG solution. Substrate: glass slide. placed substrate. The (PPG + 7.5% w/w silica and glass slide) and (0.2% w/w xanthan solution and mica) systems were completely wetting systems. The (PPG + 10% w/w and glass slide) and (1% w/w cmc solution and glass slide) were partially wetting systems. Images of the deposited drops were recorded at 60 frames s-1 using a CCD camera and were sent to a computer for storage and processing. The contact angle θ(t) and wetting radius R(t) as function of time t were acquired by analyzing each frame utilizing nonspherical fitting. Fitting R(t) data as function of time t and differentiating R(t) with respect to t estimates the velocity of the contact line, U(t). At least four identical tests were conducted for each experimental condition to ensure data reproducibility. Figure 3 presents four independent wetting tests using 7.5% w/w PPG + 10 nm silica on the glass slide conducted on different days. Data reproducibility was within 1%, a rate generally obtained for all tests. Tests with a
Table 1. Properties of Liquids Tested in This Work and Those in Reference 20 liquid
type
σ (mN m-1)
µ (Pa s) or k (Pa sn)
n
θ0 (deg)
solid substrate
PPG 1 wt % cmc 0.2 wt % xanthan 7.5 wt % PPG + 10 nm silica 10 wt % PPG + 15 nm silica PDMS + 15 nm silica20 acrylic typographic ink20
Newtonian shear-thinning shear-thinning shear-thickening shear-thickening shear-thinning shear-thinning
32.37 39.02 48.41 32.54 46.99 21.2 28.0
0.1158 7.2341 0.2847 0.0711 0.0705 9.64 16.80
1 0.5088 0.4378 1.3031 1.7282 0.79 0.62
0 20.2 0 0 26.2 0 21.6
glass slide glass slide mica glass slide glass slide glass slide glass slide
8046 Langmuir, Vol. 23, No. 15, 2007
Figure 5. Dynamic contact angle of the 0.2% w/w xanthan solution. Shear-thinning fluid and completely wetting substrate (glass slide). Present model: eq 24a.
Wang et al.
Figure 8. Dynamic contact angle of the 10% w/w 15 nm silica PPG solution. Shear-thickening fluid and partially wetting substrate (glass slide). Present model: eq 26b.
Figure 6. Dynamic contact angle of the 7.5% w/w 10 nm silica PPG solution. Shear-thickening fluid and completely wetting substrate (glass slide). Present model: eq 24b.
Figure 9. Dynamic contact angle of (a) the PDMS + silica solution and (b) acrylic typographic ink from data by Carre´ and Eustache20 were compared with the present model: eqs 24a and 26a.
Figure 7. Dynamic contact angle of the 1% w/w cmc solution. Shear-thinning fluid and partially wetting substrate (glass slide). Present model: eq 26a.
noncircular, final equilibrium drop were disregarded. Only the data with symmetrical spreading with no hysteresis were discussed further. 3.2. Results: Newtonian Fluid. Figure 4 presents the spreading test for pure PPG solution on the glass slide. The drop radius closely followed Tanner’s law, R ≈ t1/10 (Figure 4a), which is valid for Newtonian fluid in the capillary spreading regime. Moreover, the
Dynamics, Contact Angle of Non-Newtonian Fluids
Langmuir, Vol. 23, No. 15, 2007 8047
dynamic contact angle also corresponded closely with the HoffmanVoinov-Tanner law (Figure 4b)19
θ ) cTCa 3
(25)
where cT is a proportional coefficient and Ca ) (µU)/σ is the capillary number. Hence, the experimental apparatus generated quality results for Newtonian fluid in a capillary regime, partially justifying data quality in subsequent tests. 3.3. Results: Non-Newtonian Fluids. Figures 5 and 6 present the θ-U(t) data for the completely wetting systems examined: shearthinning 0.2% w/w xanthan solution and mica; and shear-thickening PPG + 7.5% w/w silica and a glass slide. Model predictions using eqs 24a and 24b agreed satisfactorily with experimental results, suggesting that the proposed model describes the wetting dynamics for both shear-thinning and shear-thickening fluids on a completely wetted surface. The lubrication approximation used for deriving eqs 24a and 24b is strictly valid at a small contact angle limit ( 1 (26b) (n + 2) (n - 1)n where θ0 is the nonzero static contact angle. This study investigated whether eqs 26a and 26b can be applied to describe partially wetting power-law systems. Figures 7 and 8 present the θ-U(t) data for the two partially wetting systems studied: a shear-thinning 1% w/w cmc solution and a glass slide and a shear-thickening PPG + 10% w/w solution and a glass slide. Model predictions using eqs 26a and 26b agreed satisfactorily with experimental results, suggesting that the modified model based on Hoffman’s proposal described the wetting dynamics for both shear-thinning and shear-thickening fluids on a partially wetted surface. For comparison sake, the data by Carre´ and Eustache20 were compared with the present model in Figure 9. The agreement is satisfactory. Hence, the derived model described the wetting dynamics of shearthinning and shear-thickening fluids on completely or partially wetted systems. n-1
n
Acknowledgment. This research is partially supported by the National Natural Science Foundation of China (grant no. 50636020). LA0701125
