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A computer model for wetting hysteresis 1. A virtual wetting balance

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Colloids and Surfaces
i koms
SURF CES
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ELSEVIER
A: Physicochemical and Engineering Aspects 89 1994) 117-l 3 1
A computer model for wetting hysteresis 1. A virtual wetting balance
Anant D. Mahalea*b, Sheldon P. Wessonb**
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‘Department of Chemical Engineering, Princeton University, Princeton, NJ USA bTRI/Princeton, 601 Prospect Avenue, POB 625, Princeton, NJ08542, USA
Received 22 November
1993;
accepted
8
December 1993
Abstract
The modified Wilhelmy plate method is widely used to determine the wettability of solid surfaces by probe liquids. This is accomplished by monitoring the wetting force experienced by a solid suspended from a microbalance during immersion into and emersion from the fluid. In the absence of surface roughness, fluctuations in force observed as the liquid slides over the solid and hysteresis between the advancing and receding modes of measurement are caused by surface chemical heterogeneity. An independent analysis based upon the fundamental rules of stick-slip behavior of the moving contact line is developed to explain these effects quantitatively and to predict the hysteresis loop for a known surface. The extent of hysteresis and the amplitude and frequency of force fluctuations are shown to depend upon the nature of the heterogeneity and its spatial distribution. A detailed description of the model is given, followed by results of computer simulations performed on model surfaces. The results agree well with previous experimental work.
Keywords:
Contact angle; Hysteresis; Surface heterogeneity; Wetting; Wilhelmy method
1 Modeling the wetting force scanning experiment
1 .I. Introduction
The acquisition of meaningful results by wetting experiments has been complicated by physical irregularities and the presence of chemical contami- nants that are invariably found on the surface of a solid. The presence of surface heterogeneities affects the apparent contact angle measured during the experiments, and hysteresis observed between the advancing and receding measurement modes has been attributed largely to these heterogeneities [ 1,2]. From a thermodynamic standpoint, contact angle hysteresis should not exist if the surface being
*Corresponding author.
studied is completely smooth and chemically homogeneous, provided that the advancing and receding measurements are done at a sufficiently slow velocity so as to remain as close to equilibrium as possible. Characteristics of wetting such as hysteresis and the kinetics of spreading could be used to advan- tage in characterizing surfaces once their measure- ment is well controlled and their interpretation understood. Traditional spectroscopic methods (e.g. X-ray photoelectron spectroscopy, polarized infrared external reflectance spectroscopy and optical ellipsometry) might then be replaced by measurements of surface wetting properties using standard probe liquids. The wetting experiment has the twofold advantage of being a simple and comparatively inexpensive measurement and of being broadly relevant technologically. Recent
0927-7757/94/ 07.00 0 1994 Elsevier Science B.V. All rights reserved
SSDI
0927-7757(94)02805-3
118
A. D. Mahale, S. P. WessonlColloids Surfaces A: Physicochem. Eng. Aspects 89 (I 994) I 17-131
research in this field, however, leaves a lot to be rules of stick-slip behavior of the moving contact understood [ 3,4]. line and the laws of mechanics. A theoretical analysis of the Wilhelmy plate scanning experiment has been attempted here. It explains the mechanism of motion of a three-phase interface over a chemically heterogeneous surface based upon an extension of the well-understood stick-slip behavior. A thermodynamic treatment of the problem along with the application of basic laws of mechanics enables the prediction of the force that is measured in a Wilhelmy measurement. On the basis of the ideas developed here, it should be possible to generate an accurate representation of a surface from wetting measurements using known probe liquids. We have done some prelimi- nary experiments to illustrate interesting features of wetting measurements that are indicative of solid surface characteristics. Further quantitative examinations of such features are needed to verify the theory. 1.3.
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efinitions
Some terms that will occur frequently in this analysis are defined here and in Fig. 1.
1) Classijication of heterogeneities based on their surface energies
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W
high energy surface or a more wettable zone; nw low energy surface or a less wettable (non- wetting) zone; t boundary separating the two types of site (2)
Length scales of interest on the surface of the Wilhelmy plate
1.2.
Theory
A modified Wilhelmy plate approach is com- monly adopted to evaluate the wettability of films by probe liquids [ 51. This is done by monitoring the wetting force experienced by a film which is suspended from a microbalance while being immersed into or pulled out of a liquid. Two important features of a typical experiment are the continuous fluctuations in force observed as the liquid slides over the film, and the hysteresis between the advancing and receding modes of measurement. These are a result of one or several perturbing factors of a physical nature, such as surface roughness [ 63, molecular orientation [ 71, chemical heterogeneity of the solid surface, or swelling of the solid by diffusion of the wetting liquid.
L wx
width of a high energy site measured in a direction perpendicular to the direction of motion of the surface
L nwx
width of a low energy site measured in a direction perpendicular to the direction of motion of the surface
L wy
length of a high energy site measured in the direction of motion of the surface
L nwy
length of a low energy site measured in the direction of motion of the surface
L tx
total length of surface perpendicular to the direction of motion Lt, total length of film in the direction of motion (3)
Length scales based on the equilibrium shape of the meniscus formed by a probe liquid on the surface
The work reported here deals with surface chem- ical heterogeneities. If the effects of such heteroge- neities were understood, a quantitative estimate of the degree, type and distribution of surface perturb- ations could be obtained from wetting force scans. Since previous work is not sufficient in its applica- tion to real surfaces, we have attempted an inde- pendent approach based upon the fundamental
H,
height of contact line relative to the free surface of the liquid (fs) in the case of a w zone
H, < 0
if 0, > 90”) where 0 is the contact angle)
H nw
height of contact line relative to the free surface of the liquid in the case of an nw zone
H,,<O
if 8,,>90”)
J
the “jump span”, equal to the absolute differ- ence between
H,
and
H,,
A. D. Mahale, S. P. WessonlCoiloids urfaces A: Physicochem. Eng. Aspects 89 1994) I1 7-131 119
Lnwy
LWY
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nw
Fig. 1. Definitions of some terms used in our analysis of the wetting force scanning experiment. FS denotes free surface.
4) System constant aficting the values of H, and H nw Ho = 2rlpg)“2
where p is the density of the probe liquid, y is the surface tension of the probe liquid, and g is the gravitational constant, taken as 981.0 cm sp2. (5)
Relevant surface forces used in the analysis
fw
force per unit distance along the contact line exhibited by the w zone
f nw
force per unit distance along the contact line exhibited by the nw zone F, total surface force (6)
Diflerent mechanisms of motion of the contact line and the free surface of the liquid
Stick/Pin (p) the contact line remains stationary, and the free surface moves Slip (sp) the contact line rapidly traverses a certain distance, and the free sur- face remains stationary Slide (sd) the contact line and the free surface move together
1.4. Thermodynamic derivation: force during a Wilhelmy experiment
Consider a system which includes a solid(s), a liquid(l) and its container (c), as shown in Fig. 2. Using the definition of surface tension y for a process at constant temperature and pressure, the work done by the system minus the volumetric
;
Fig. 2. Arrangement for the Wilhelmy experiment.
12
A. D. Mahale, S. P. WessonlColloidr Surfaces A: Physicochem. Eng. Aspects 89 1994) 117-131
work P dV is given by [S]
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-6W=y6A 1) Now
imagine a process in which the solid is vertically displaced a distance dy from y as shown in Fig. 2. The free surface moves by the same amount. In this case 6
W = F
dy -
m,g
dy +
ApgA,y
dy (2) where
A,
is the cross-sectional area of the solid, Ap = (p, - p,) is the difference between the densities of the liquid and vapor (v) phases, and
m,g = p, - p,)A,hg
is the weight of solid in the vapor phase. The total surface energy change occurring in this step is given by Y dA = ~1s A,, + ysv dA,, + ~1” dA1, + yci d&i + ycv dA,, (3) Since the cross-sectional area of the solid is much smaller than that of the container
dA,, =
dA,, = 0 For a solid with constant perimeter, the liquid- vapor surface area does not change as the solid is immersed in the liquid, i.e. dA,, = 0 so that Y dA = ~1s Ai, + 7s” dA,, (4) Applying Young’s force balance for equilibrium Y dA = (~1, ysv) dA,, = - (riv cos 0) d& Combining Eqs. (l), (2) and (5) gives (5)
F = m,g - ApgA,y + ylv cos B)P 6)
where P is the perimeter of the solid. The microbalance reading during a Wilhelmy plate experiment measures the sum of the vertical component of the surface force and the weight of the plate corrected for buoyancy. In Eq. (6) the total surface force is given by
F, = ~1,
COS 8)P = CfiLix
(7) wherefi is the zonal force per unit distance along the contact line due to zone i, ylV is the surface tension of the liquid with respect to the vapor, 8i is the equilibrium contact angle made by the liquid with the solid surface i, L, is the width of zone
i,
and the summation is over all zones along the contact line at any position of the free surface of the liquid. During a wetting experiment on a strip, the contact line follows one of the following three states of motion. (1)
Slip mode.
In this case, the free surface is stationary, but the contact line travels rapidly over the surface. The meniscus changes shape, the observed contact angle thus changing continu- ously, as does the measured force, from its initial value to the next stable value. The force measured by the microbalance is given by Eq. (6), and the wetting force is obtained after subtracting the weight of the plate and correcting for buoyancy effects. (2)
Stick/pin mode.
In this case, the contact line does not move but the free surface moves. The wetting force contribution is calculated from Eq. (6). Here again, there is a change in the shape of the meniscus caused by the motion of the free surface. (3)
Sliding mode.
This involves motion of both the contact line and the free surface. The wetting force can be obtained from Eq. (6) after subtracting the weight of the plate from the total force and correcting for buoyancy effects.
1.5.
Wetting force analysis for a heterogeneous strip - one-dimensional model
The object under consideration here is shown in Fig. 3(a). If, as in the case of a typical dynamic Wilhelmy experiment, the advancing and receding modes are carried out sufficiently slowly, any stable meniscus shape observed is an equilibrium shape. The motion of the liquid contact line on the surface occurs in jumps; this has been referred to as the stick-slip mechanism. During liquid advance, the following set of events occurs.
A. D. Mahale, S. P. WessonlColloids Surfaces A: Physicochem. Eng. Aspects 89 1994) I1 7-131 121
a)
F - weight
9
Advancing mode Receding mode Fig. 3. One-dimensional model: wetting force analysis for a strip with surface heterogeneities. (a) The system. (b) Distances traversed by the contact line and the free surface in the advancing mode. (c) Distances traversed by the contact line and the free surface in the receding mode.
(1) The contact line pins at the wetting-to-non- wetting transition. (2) The contact line slides over the entire non- wetting zone. (3) Slip occurs at the non-wetting-to-wetting transition. The length traveled by the contact line during slip and the distance traversed by the free surface during pinning are shown in Fig. 3(b). The contact line slips a distance equal to the jump span on the wetting zone. If the length of this region is less than the jump span, the contact line will slip over the entire wetting zone and pin at its terminal location. Pinning in this case occurs over a free surface displacement equal to the size of the wetting zone. However, if the size of the wetting region is greater than the jump span, after the slipping mechanism, the contact line will slide over the

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