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Wetting Dynamics of Alkyl Ketene Dimer on Cellulosic Model Surfaces Gil Garnier,* Marylise Bertin, and Miroslava Smrckova Paprican and Department of Chemical Engineering, Pulp and Paper Research Centre, McGill University, Montreal, Quebec, Canada H3A 2A7 Received March 12, 1999. In Final Form: June 7, 1999 The dynamic wetting of a commercial alkyl ketene dimer (AKD) wax was measured on model cellulosic surfaces. The variables investigated were temperature and the surface composition. The model surfaces consisted of cellulose and cellulose acetate films as well as glass. These surfaces are smooth by industrial standards but not on a molecular level. The objective of the study was to predict the extent of AKD wetting during the time frame of papermaking. For smooth surfaces, AKD particles wet but do not spread on the hydrophilic surfaces investigated. AKD wetting proceeds from the balance of the interfacial forces with the viscous dissipation. The effect of gravity can be neglected for papermaking conditions. The HoffmanTanner equation modified for partial wetting provided a very good fit of the dynamic wetting. The slope of the graph is a function of temperature but not of the solid surface composition. Maslyiah’s model also fits the experimental results well, but with a physically unrealistic value of the fitting parameter. For partial wetting, the complex but rigorous Cox equation is recommended to estimate the slip length over macroscopic wetting dimensions.
Introduction Papermaking is a fast industrial chemical process with production velocities well over 25 m/s. The paper industry involves many unit operations controlled by surface and colloidal engineering. Among these, the dynamics of wetting is believed to play a critical role during internal sizing as colloids, previously adsorbed onto the fibers, are heated and allowed to wet during the drying operations. It is desirable to predict the extent of wetting or, conversely, to calculate the characteristic time required to reach a precise extent of wetting. The understanding of the dynamics of wetting has drastically progressed in the past decade.1-5 Many models derived from first principles are widely used.6,7 These models follow two basic approaches. The first considers wetting as a fluid mechanic process in which one fluid displaces another. Navier-Stokes equations can then be solved for each fluid, with certain assumptions, resulting e.g. in the Cox equation.3 The second relies on a molecular vision as force balances are performed on a population of molecules. The principal hypothesis is that the motion of the three-phase line is determined by the statistical kinetics of molecular events within the three-phase zone. The model of Blake is a well-known example.8,9 De Coninck et al. modeled up to 200 000 molecules and even considered the effect of surface heterogeneity.10-12 Both types of * Corresponding author. (1) De Gennes, P. G. Dynamics of wetting. In Liquids at Interfaces; Charvolin, J., Joanny, J. F., Zinn-Justin, Eds.; North-Holland: Amsterdam, 1990; pp 273-291. (2) De Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827. (3) Cox, R. G. J. Fluid Mech. 1985, 168, 169-194. (4) Joanny, J. F. J. Appl. Phys. (Special Issue) 1986, 249. (5) Kistler, S. F. Hydrodynamics of Wetting. In Wettability; Berg, J. C., Ed.; Dekker: New York, 1993; pp 411-429. (6) Fermigier, M.; Jenffer, P. J. Colloid Interface Sci. 1991, 146, 226241. (7) Mumley, T. E.; Radke, C. J.; William, M. C. J. Colloid Interface Sci. 1986, 109, 398-412. (8) Blake, T. D. In Wettability; Berg, J. C., Ed.; Dekker: New York, 1993; p 251. (9) Blake, T. D.; Clark, A.; De Coninck, J.; de Ruijter, M. J. Langmuir 1997, 13, 2164-2166. (10) Ruijter, M. J.; De Coninck, J. Langmuir 1997, 13, 7293-7298.
models efficiently describe the dynamic wetting of monodisperse, Newtonian fluids such as silicone oil on molecularly smooth surfaces. Much less known is the susceptibility of these models to chemical and physical heterogeneity. The predictions of theory for the effects of temperature and surface energy on the dynamics of wetting have seldom been compared to experimental results. This article aims at modeling the dynamics of wetting of a wax, AKD, on different model surfaces as a function of temperature. Alkyl ketene dimer (AKD) is widely used in the paper industry to increase the paper hydrophobicity and improve its printability. This is typically done by adding a polyelectrolyte stabilized AKD emulsion directly to the paper furnish prior to the paper machine. AKD sizing13-15 is believed to occur by a three step mechanism involving (1) the retention of the AKD particles (0.5-1 µm) on the pulp fibers, (2) the spreading of the AKD particle to a monolayer thickness film, and (3) some reconformation involving a covalent cellulose-AKD bond. The first16,17 and last18 steps are now well-understood. Although AKD spreading is commonly admitted, it has never been observed nor quantified. Indeed, recent studies in our laboratory proved that AKD can only partially wet smooth cellulose surfaces, i.e., the contact angle θE > 0; however, they do not spread, i.e., θE ) 0.19,20 (11) Cazabat, A. M.; Valignat, M. P.; Villette, S.; De Coninck, J.; Louche, F. Langmuir 1997, 13, 4754-4757. (12) Urban, D.; De Coninck, J. Phys. Rev. Lett. 1996, 76 (23), 43884391. (13) Scott, W. E. Principles of Wet End Chemistry; Tappi Press: Atlanta, GA, 1996; pp 99-109. (14) Lindstrom, T.; Eklund, D. Paper Chemistry: An Introduction; DT Paper Science Publications: 1991; pp 192-222. (15) Roberts, J. C. In Paper Chemistry; Roberts, J. C., Ed.; Chapman and Hall: London, 1989; pp 114-131. (16) Lindstrom, T.; Soderberg, G. Nord. Pulp Pap. Res. J. 1986, 1, 26-33. (17) Lindstrom, T.; O’Brian, H. Nord. Pulp Pap. Res. J. 1986, 2, 3138. (18) Yu, L.; Garnier, G. 11th Fundamental Research Symposium; Cambridge, U.K.; 1997; Vol. 2, pp 1021-1046. (19) Garnier, G.; Wright, J.; Godbout, L.; Yu, L. Colloids Surf. A 1998, 145 (1-3), 153-166.
10.1021/la990297i CCC: $18.00 © 1999 American Chemical Society Published on Web 08/20/1999
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sure (Πe) is zero,
γSV ) γ cos θE + γSL
(2)
Combining eqs 1 and 2 yields, for partial wetting,
S ) γ(cos θE - 1)
(3)
A precursor film is known to extend a few millimeters ahead of the drop for complete wetting.21 By minimizing the free energy of the film, the crossover thickness from the wedge to the precursor film (eo) can be shown to vary as Figure 1. Schematic representation of spreading and partially wetting droplets.
The wetting dynamics of a commercial AKD wax over a variety of model surfaces is presented in this study. Model surfaces were selected to minimize surface roughness and to eliminate their influence from the study. These surfaces consist of cellulose film, cellulose acetate film, and glass. Cellulose is the model representing pulp fibers, which are too rough for accurate dynamic wetting studies. Cellulose acetate and glass have different hydroxyl concentrations and therefore different surface energies than cellulose. These surfaces are smooth for industrial standards but not on a molecular level. A commercial AKD wax was selected and used for these experiments to investigate the wetting phenomena relevant to the paper industry.
Many comprehensive analyses of wetting can be found in the literature.1-5 This section reviews the elegant development of P. G. de Gennes.1,2 Wetting can be analyzed at equilibrium from a macroscopic thermodynamic point of view or from fluxes and force balances to quantify its dynamics. (1) Wetting at Equilibrium. Wetting presents two basic cases at equilibrium: partial wetting and complete wetting. Figure 1 schematically represents the moving wedge of spreading and wetting droplets. In complete wetting, the liquid droplet spreads onto a surface to reach an equilibrium angle of zero degrees (θE ) 0), whereas for partial wetting the droplet reaches θE > 0. No wetting occurs if θE g 90°. The wetting behavior of a system at equilibrium can be quantified with the spreading coefficient (S) defined as
(1)
where γij is the interfacial tension between two phases represented by S for the solid, L for the liquid, and V for the vapor. For clarity γLV is simply denoted as γ. The spreading coefficient is a measure of the difference in surface energy between the dry solid and the moist solid covered by a macroscopic film of liquid. Complete spreading happens for S g 0, while for S < 0 partial wetting is achieved. The Young equation relates θE to the surface energies. From a force balance at the triple phase line of a droplet in equilibrium, provided the equilibrium spreading pres(20) Garnier, G.; Yu, L. J. Pulp Pap. Sci. 1999, 25 (7), 235-242.
1/2
(4)
where a is of molecular dimensions defined by a2 ) A/6πγ, with A being the Hamaker constant. The film thickness is a decreasing function of S. (2) Dynamics of Wetting. The dynamics of wetting results from the balance of interfacial forces, gravity, viscous forces, and inertial forces. The former two are driving forces while the latter two are forces opposing the movement. The relative importance of the forces is best expressed with three dimensionless numbers, namely, the Bond number (Bo), the capillary number (Ca), and the Weber number (We) defined as
Bo )
FgL2 gravity force ) interfacial force γ
(5)
viscous force µU ) interfacial force γ
(6)
FU2L inertial force ) interfacial force γ
(7)
Ca )
Theory
S ) γSV - γSL - γ
3γ (2S )
eo ) a
We )
where U ) dR/dt is the velocity of the macroscopic threephase line and L denotes some characteristic length, typically the drop radius (R). µ and F represent the liquid viscosity and density, respectively, and g is the gravitational acceleration. For small droplets, gravity and inertial forces are usually negligible and wetting proceeds from the ratio of the viscous and interfacial forces. For such systems Hoffman first reported that the apparent contact angle can be correlated as a function of the capillary number by a sigmoid master curve. From Hoffman’s experimental results, the following relationship can be extrapolated to describe the dynamics of wetting:22
θ3 ∝ Ca
(8)
which is commonly known as the Hoffman-Tanner equation. For partial wetting it is usual to compare wetting under a constant driving force and to express eq 8 as23,24
(θ3 - θE3) ∝ Ca
(9)
The precursor film height profile can be derived from the Navier-Stokes equation by applying the lubrication approximation (low Re). The profile of the precursor film, (21) Villette, S.; Valignat, M. P.; et al. Langmuir 1996, 12, 825-830. (22) Hoffman, R. L. J. Colloid Interface Sci. 1975, 50, 228-241. (23) Seaver, A. E.; Berg, J. J. Appl. Polym. Sci. 1994, 52, 431-435. (24) Basu, S.; Nandakumar, K.; Masliyah, J. H. J. Colloid Interface Sci. 1996, 182, 82-94.
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f(x), decays very slowly accordingly to
f(x) )
(
a2 1 θ3 xo - x
)
(10)
eo ∝ a/θ
(11)
where xo is an integration constant. Equations 10 and 11 explain why the precursor films can only be seen under conditions of complete wetting for which θ is small and eo is large. Strong liquid-solid interactions tend to stabilize thick films. Dissipation arises because of the viscous friction in the spreading film. The total viscous dissipation per unit length of contact line, DV, can be expressed as
DV )
3ηU2 ln|xmax/xmin| θ
(12)
where xmax and xmin are integration constants corresponding to the boundary conditions of the system. xmax can be related to the macroscopic size of the droplet (R), while xmin is on the order of the angstrom. We can compare de Gennes’ calculations to the rigorous hydrodynamic approach of Cox by defining two ratios:2,3
1 ) xmin/xmax ≈ s/R
(13a)
2 ) s/eo
(13b)
s is the distance over which slippage occurs to avoid the singularity at the three-phases line. The total viscous dissipation is related to the driving force, FD, by
DV ) FDU
(14)
The driving force for the spreading process results from the imbalance of forces in the Young equation:
FD ) γSV - γSL - γ cos θ
(15)
FD ) γ(cos θE - cos θ)
FD ) S + γ(1 - cos θ) ≈ S +
γθ2 2
(16)
The first term of the right-hand side (S) is typically 3 orders of magnitude higher than the second term.1,2 Assuming low contact angles (θ , 1) yields the second approximation of eq 16. Combining eqs 14 and 16
DV ) (S + 1/2γθ2)U
(17)
Most of the viscous dissipation occurs in the precursor film (S), while the second term represents an angular correction. Neglecting S1,2 and substituting equation result in
ηU2 γ 3 ∼ θ γθ3 η
(18)
Equation 18 is simply a form of the Hoffman-Tanner equation (eq 8). This is an equation of state describing the wetting dynamics of all spreading liquids.1,22,25,26 Two powers of θ come from the driving force and one from the viscous dissipation. This equation implies that the wetting (25) Tanner, L. H. J. Phys. D: Appl. Phys. 1979, 12, 1473-1484. (26) Lelah, M. D.; Marmur, A. J. Colloid Interface Sci. 1981, 518525.
(19)
Comparing eqs 19 and 16 confirms that the driving force for partial wetting is always smaller than that for complete wetting. The assumption of low contact angle (θ , 1) does not hold for most of the process, and the common simplification of V ∼ hθ3 relating droplet height, h, with its contact angle becomes erroneous. Assuming the drop to remain a spherical cap shape throughout the wetting process, it can be shown geometrically that
V ) 1/6πh(3R2 + h2) ) 4/3πro3 cos θ )
(20)
1 - (h/R)2
(21)
1 + (h/R)2
where ro is the radius of the spherical droplet prior to wetting. Basu et al. investigated this approach to model the dynamics of dewetting of bitumen films onto droplets.24 They expressed the variation of contact angle with time or capillary number as
(
)([
]
(1 - cos θ) 2 θ sin θ dθ ) 3+ / dt 3µR sin θ 1 - cos θ 2 1 γ(cos θE - cos θ) (22) 1+ sin θ ln( -1)
)] )
[ (
[
R ) (4/3)1/3ro/
which can be written as
U≈
rate (U) is independent of the spreading coefficient or the driving force. De Gennes referred to this phenomenon as the compensation theorem: driving force and viscous dissipation are similar but opposite functions of the coefficient of spreading (S).1,2 A system with small S has a small driving force for spreading but also a small viscous dissipation term, whereas the higher driving force of a system with large S also has a large viscous dissipation. For partial wetting (S < 0), there is no precursor film and the driving force becomes
1
1 - cos θ (1 - cos θ) + 2 sin θ 6 sin3 θ
]
3 1/3
3 ln(1-1) dθ ) dCa [(cos θE - cos θ) + θ sin θ]
(23)
(24)
Experimental Section (1) Materials. A cellulose acetate of low molecular weight (19 000) from Eastman Kodak with a 39.8% acetyl content was used as received. All solvents were of HPLC grade. The microscope glass slides (22 mm × 50 mm; thickness 2) were purchased from Fisher Scientific Co. A commercial AKD wax (Raisio Chemical, North America) and made from a fatty acid mixture of C14 to C20 with C18 as the main component was recrystallized three times in hexane. For the wetting experiments, the AKD wax was extracted to remove unreacted fatty acids that would otherwise behave as surfactant. Figure 2 presents the chemical structure of AKD and its reactional pathway with cellulose and water. AKD viscosity and surface tension were measured with a Brookfield rheometer (spindle 1) and a Sigma 70 KSV Wilhelmy balance, respectively. Table 1 presents the effect of shear on viscosity. The film preparation was previously described.18 Basically, the glass slides were cleaned with nitric acid (70%), rinsed with ultrapure water, and dried at 120 °C. Cellulose acetate films were prepared by slowly dipping the glass slide into a 5% (w/v) solution of cellulose acetate in acetone using a Wilhelmy balance and air-dried. These cellulose acetate films can be regenerated into cellulose by soaking in a 0.5% sodium methoxide solution overnight. The regenerated films are then washed with water and methanol, air-dried, and stored in a desiccator at 24 °C and
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Garnier et al. Table 2. Physical Properties of the AKD Molten Wax as a Function of Temperature temp (°C)
surface tension (m N/m)
viscosity (m Pa‚s)
density (g/mL)
75 65 60
27.0 28.8 29.4
1.80 2.36 2.69
0.852 0.860 0.865
Table 3. Dimensionless Number Range Covered by the Wetting Experiments temp (°C) 75 65
Bo
Ca
We
8.31 × to 1.3 × to 1.2 × 10-11 8.31 × 10-4 1.2 × 10-4 to 6.2 × 10-6 1 × 10-7 to 2.8 × 10-10 10-4
10-4
10-6
10-7
speed
viscosity (mPa‚s)
speed
viscosity (mPa‚s)
speed
viscosity (mPa‚s)
later analyzed with ImagePro Analysis Software (version 3.01). A partial reflection or shadow is necessary to determine the position of both left and right apexes. The contact angle was measured from the mean of the angles at the apexes divided by 2. The radius (R) and height (h) of the drop were measured as half the diameter and the height of the system drop/reflection. The sampling time was attributed to each analyzed image from the recording rate of the camera. Zero time is defined as the moment when the drop first touches the surface.
6 12
1.56 1.64
30 50
1.73 1.80
60 100
1.83 2.00
Results
Figure 2. AKD chemical structure and reaction pathway with cellulose and water. Table 1. Effect of Shear on the AKD Viscosity. Brookfield Rheometer Using Spindle No. 1 at 75 °C
34% relative humidity. The root mean square of the surface roughness, as measured by atomic force microscopy, was on the order of 10 nm for the cellulose films. (2) Instrumentation and Methods. The contact angle was measured on a video contact angle system built in our laboratory. It comprises (Figure 3) an environmental chamber and an elevated temperature attachment (Rame´- Hart) and two temperature controllers all placed on an antivibration table (Cornwall). A glass microsyringe was used with syringe tips (no. 28) with an outer diameter of 0.364 mm. The glass piston of the syringe was replaced by a Teflon piston to prevent leaks. The droplet average initial diameter was 1.5 mm. The images were captured with a Sony Hi Resolution video camera and were recorded on a Sony SVHS HI-FI video recorder. The images were
The physical properties of AKD influencing the dynamics of wetting are presented at different temperatures in Table 2. While density is relatively constant, surface tension and especially viscosity exhibit a nonnegligible temperature dependence over the range investigated. Viscosity experiments showed that AKD can be considered a Newtonian fluid over a small range of temperature and shear (Table 1). The melting point varies from 50 to 51 °C, which reflects some chemical heterogeneity of the AKD wax. A typical AKD droplet wetting a cellulose film is shown in Figure 4. The height (h), radius (R), and contact angle of both apexes (θ) are followed as a function of time, and the velocity of the macroscopic wetting line can be
Figure 3. Experimental apparatus for the dynamic wetting study.
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Figure 4. Picture of a typical AKD droplet at equilibrium on a cellulose film.
Figure 6. AKD dynamic wetting on different surfaces at 65 °C. Table 4. Effect of the Surface on the Equilibrium Contact Angle (θE) of AKD at 65 °C surface
θE (deg)
time to reach θE (s)
Sa (m N/m)
glass cellulose acetate cellulose
17.5 12.0 10.3
1.8 3.9 4.1
-1.3 -0.5 -0.6
a
Calculated from eq 3.
Figure 5. Comparison of the experimental droplet height with the theoretical assuming spherical cap.
calculated. The Bond number (Bo), Capillary number (Ca) and Weber number (We) can then be computed from the limiting conditions (Table 3). In all instances, the Weber number is at least 3 orders of magnitude smaller than Ca and Bo. Because of the critical values of We, the effect of inertial force can be neglected. As expected for a small droplet, the Bond number is significantly smaller than 1; the usual assumption of neglecting gravity can reasonably be made. In the absence of gravity, droplets are then assumed to retain their spherical cap shape throughout wetting and to be described by eqs 20 and 21. The predicted droplet heights are compared to those measured in Figure 5. The good linearity confirms the validity of neglecting gravity. The wetting dynamics can therefore accurately be modeled in terms of capillary number, resulting from the ratio of viscous to interfacial forces. Once the driving forces for wetting have been identified, the type of wetting must be confirmed. Previous experiments on the equilibrium wetting of sessile droplets suggested a partial wetting behavior.19,20 Furthermore no foot or precursor film was observed by atomic force microscopy of the droplets.20 This was however expected since the critical thickness of the film (Å) lies below the sensitivity of the instrument (nm). Only thick precursor feet could have been seen. The wetting dynamics of AKD droplets deposited onto glass, cellulose acetate and cellulose are compared in Figure 6. The equilibrium contact angle and the time after which it was reached are presented in Table 4 at 65 °C. Spreading coefficients calculated from eq 3 are also compared. The wetting behaviors on the three surfaces are identical. The initial wetting velocities (slopes) are similar but level-off to different equilibrium contact angles (θE). The equilibrium angle of AKD on glass is the highest, while those on cellulose acetate and cellulose are similar. This is as expected because the surface energy of the hydrophobic AKD on glass (γ(AKD-glass)) is higher than that of AKD on the hydrophobic cellulose acetate (γ(AKDcellulose acetate)). Surface roughness is not expected to
Figure 7. AKD dynamic wetting on glass at different temperatures.
play any significant role in the wetting dynamics since atomic force microscopy revealed smooth and identical surfaces with a root mean square typically of 10 nm. For our experiments, the spreading coefficients are only weakly affected by the surface composition as the equilibrium contact angles are small and relatively similar. The influence of temperature on the wetting dynamics of AKD on glass was also studied at 60, 65, and 75 °C (Figure 7). The curves have similar profiles that can be divided into two regions: a fast decrease of contact angle during the first second and a plateau leading to the equilibrium contact angle (θE). The equilibrium contact angle and spreading coefficients are shown in Table 5. Temperature significantly affects the spreading coefficient with the combined effect of the temperature on θE and γ. A peculiar behavior was observed at 60 °C. After the onset of fast wetting (1.5 s), the AKD droplet retracted by about 5°. This phenomenon was completely reproducible and well within the experimental error ((0.2°). To our knowledge this observation can have two explanations: (1) the domination of an elastic flow component, promoted
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Table 5. Effect of Temperature on the Equilibrium Contact Angle (θE) of AKD on Glass temp (°C)
θE (deg)
time to reach θE (s)
Sa (m N/m)
60 65 75
26.0 17.5 14.6
2.6 1.8 1.2
-3.0 -1.3 -0.9
a
Calculated from eq 3.
Figure 9. Hoffman-Tanner plot modified for partial wetting showing the dynamic wetting of AKD on glass at different temperatures.
Figure 8. Hoffman-Tanner plot modified for partial wetting showing the dynamic wetting of AKD on different surfaces at 65 °C.
by the closeness to the melting point, over the viscous flow component and (2) the competition between AKD vapor deposition with AKD wetting. The elastic character of AKD was obvious in the temperatures ranging from 55 to 60 °C as the droplets were rebounding on the surface immediately after deposition (t < 0.2 s). This often leads to a chaotic wetting behavior resulting in the simultaneous translation and wetting of the droplet. The reproducibility of the phenomenon and the time of retraction (2.5 s) raise doubts about the first hypothesis. The second possibility deserves more attention. As an AKD droplet wets a surface, it also reaches an equilibrium with its vapor phase. The AKD vapor can then adsorb on the surface to be wetted, therefore modifying γSV. This phenomenon was studied elsewhere for AKD19 and is well-known for volatile molecules and referred to as autophobicity.27-29 At 60 °C, vaporization of AKD occurs, but it is slower than at higher temperature. The low vapor pressure at 60 °C means that a longer time is required to vaporize, diffuse, and adsorb a sufficient number of AKD molecules to affect the wetting process. At higher temperatures, as more AKD evaporates, this mechanism might happen within the first second and therefore go unnoticed. Discussion The wetting kinetics can best be analyzed in terms of capillary numbers. The driving force, expressed as (θ3 θE3), can be related to the capillary number and compared to the linear relationship predicted by the modified Hoffman-Tanner eq 9. Figure 8 presents the HoffmanTanner plot of AKD wetting on three different surfaces at the same temperature (65 °C). Linear plots of similar slope of 1.2 × 103 are obtained (dimensionless when the angle is in radian). This is as expected. Assuming a purely hydrodynamic behavior, the effect of the liquid-solid (27) Novotny, V. J.; Marmur, A. J. Colloid Interface Sci. 1991, 145, 355. (28) Zisman, W. A. Adv. Chem. Ser. 1964, 43. (29) Adamson, A. W.; Gast A. P. Physical Chemistry of Surfaces, 6th ed.; Wiley-Interscience: New York, 1997; p 783. (30) Oliver, J. F.; Huh, C.; Mason S. G. Colloid Surf. A 1980, 1, 79. (31) Dettre R. H.; Johnson R. E. Adv. Chem. Ser. 1964, 43.
interaction should only appear in the ordinate with θE; by subtracting it from the dynamic angle θ, the driving force is therefore normalized. Similar wetting velocities (U) should then result for a constant temperature, with γ and µ being constant. This also confirms the similar and negligible roughness of the model surfaces. A microscopic roughness of 10 nm is insufficient to pin the macroscopic wetting line. No grooves, inducing capillarity, are present on the model surfaces. The modified Tanner-Hoffman equation was also plotted for AKD wetting glass at different temperatures (Figure 9). Linear relationships also resulted. However the slope of these curves is a function of temperature. Dimensionless slopes of 1.2 × 103 and 0.8 × 103 were measured at 65 and 75 °C, respectively. This means that for a given driving force (θ3 - θE3), the wetting velocity varies with temperature. Such a result is not obvious and deserves further analysis. Temperature modifies θE, γ, and µ, which affect both the abscissa and ordinate of the graph. In a classic paper,2 de Gennes proved that, for complete wetting, the wetting velocity is independent of the spreading coefficient (S). This was explained with the compensation theorem: driving forces and viscous dissipation are similar but opposite functions of S. Most of the viscous dissipation term results from the precursor film. Furthermore, the precursor film thickness decreases as a function of S, while its length increases as reported by eqs 10 and 11. Viscous dissipation increases with the length and the decrease of the film thickness. In partial wetting, θ stabilizes quickly to its equilibrium value (θE). Because of the important θE, the crossover thickness from the wedge to the precursor film (eo) is of the molecular thickness and there is no tail to dissipate the energy by viscous forces. The viscous dissipation in the precursor film can then be considered as negligible. Most of the viscous dissipation therefore occurs within the droplet by a caterpillar-like movement of the molecules. The dependence of the macroscopic viscous dissipation upon S is unclear. The process is however expected to depend mostly on viscosity, which decreases as a function of temperature; viscous dissipation should thus decrease similarly. For partial wetting systems the driving force is described by eq 19. γ and θE are both functions of temperature. The maximum driving force is achieved in the first instance of the wetting process as (cos θE - cos θ) is at its maximum. Assuming 50° as the dynamic angle (θ), driving forces per unit lengths of 7.6, 9.0, and 9.6 mN/m are calculated at 60, 65, and 75 °C. The driving force increases with temperature. The resulting force
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for partial wetting. However, Masliyah successfully modeled bitumen dewetting with realistic values of 1 ) 0.03. This is as expected since in the first instances of dewetting, the receding film dissipates energy similarly to a precursor film ahead of a wetting droplet. Maslihah’s model is therefore valid for describing the first instances of dynamic dewetting and the dynamic wetting of spreading systems. It is, however, inappropriate to model the dynamics of partially wetting liquids, for which the more complex Cox equation is preferred.3 The practical implications of the wetting behavior of AKD in papermaking are discussed elsewhere.32
Figure 10. Comparison of the experimental dynamic wetting of AKD on glass with the predictions of Maslyiah’s model (65 °C). Dots indicate the experimental results, while the line represents the model with ln(1 - 1) ) 45.
acting on the AKD drop then increases with temperature with the increase of the driving force combined with the decrease of the viscous dissipation. Applying Newton’s second law to an AKD droplet predicts that its wetting rate increases with temperature. This expected behavior is observed in Figure 9, which shows that, for a constant driving force, the velocity of the three-phase line increases with temperature. We would also expect the three-phase wedge profile to be a function of temperature. This could be expressed with different slip lengths (s) for a given radius (R), and therefore different values as a function of temperature. Masliyah’s dewetting model was modified to model the dynamics of AKD partial wetting and to estimate the 1 ratio (eqs 22 and 23). Figure 10 compares the experimental results to those predicted by the model. The only fitting parameter is ln(1-1). Best curve fitting is achieved with a value of 1 ) 10-45. Such a low value for the ratio of the critical slip length over the macroscopic dimension of the droplet is not physically realistic, since 1 should rather range from 10-2 to 10-7.3,24 Masliyah’s model relies on eq 12 to describe the total viscous dissipation. This equation is valid only for complete wetting since it only takes into account the dissipation in the precursor film, nonexistent
Conclusion The dynamic wetting of AKD on model surfaces can be predicted from the balance of interfacial and viscous forces. The Hoffman-Tanner equation modified for partial wetting describes well the phenomenon. As predicted, the slope of the graph is independent of the surface energy of the solid as it is already included in the driving force. However, the slope is a function of temperature. This is explained by a higher force, giving the droplet a higher acceleration and therefore a higher velocity (U). This higher force balance is caused by an increased driving force (FD) combined with a lower viscous dissipation (FV) at higher temperatures. The accurate prediction of the dynamics of wetting of AKD from a balance of the interfacial and viscous forces corroborates that surface irregularities do not affect the dynamics of wetting on model surfaces. The same conclusion should hold for pulp fibers if their roughness is comparable to that of the model surfaces. Masliyah’s model, derived from de Gennes principles for dewetting, fits relatively well the experimental results. An unrealistic value of 1 ) 10-45, however, shows the shortcoming of the model in describing physically partial wetting in the absence of a precursor film. Acknowledgment. We wish to acknowledge Drs. T. G. M. van de Ven and G. V. Laivins for valuable comments and L. Godbout for technical expertise. LA990297I (32) Garnier, G.; Godbout, L. Submitted for publication in J. Pulp Pap. Sci.
