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Evaluation of the Force Required to Move a Coalesced Liquid Droplet along a Fiber Ryan Mead-Hunter,*,† Benjamin J. Mullins,†, Thomas Becker,†,‡ and Roger D. Braddock§ †
Fluid Dynamics Research Group, ‡Nanochemistry Research Institute, Curtin University, GPO Box U1987, Perth WA 6845, Australia, and §Atmospheric Environment Research Centre, Griffith University, Nathan QLD 4111, Australia Received October 15, 2010. Revised Manuscript Received November 15, 2010
This work presents a theoretical model describing the force required to move a coalesced liquid droplet along an oleophilic filter fiber. Measurements have been made using the atomic force microscope (AFM) to examine these forces over a range of fiber and droplet diameters as well as oil properties. Good agreement between measured and modeled forces was found. The influence of droplet displacement perpendicular to the fiber on the force required to move the droplet has also been determined experimentally and theoretically. It was found that fiber surface inhomogeneities are likely to influence results. This work has also established empirical relationships that can be used to predict the force, based on a known droplet volume, for the liquid types used.
Introduction
filter capture efficiency, which is modeled on the basis of a single fiber.6-8 The equations describing efficiency in this manner are collectively referred to as the single-fiber efficiency theory, which consists of an equation describing the collection efficiency for each method of droplet capture. Given the complex (random) structure of fibrous filters (as illustrated in Figure 1), this single-fiber approach provides a simple method of examining the fundamental behavior of an otherwise complicated system. Accordingly, we have chosen to adopt this approach in the study of the movement of coalesced liquid droplets along a fiber. Fibrous filters that are exposed to liquid aerosols (such as oil mists, with particle sizes typically in the range of 1-1000 nm) will collect these particles through diffusion or inertial mechanisms, which will (in the oleophilic case) initially form a very thin film on the fiber surface. This film will almost immediately be broken up into an array of axisymmetric droplets via the Plateau-Rayleigh instability.9 These large droplets will continue to grow in size with the collection of further aerosol particles until the point where they will drain from the fiber under the influence of drag and gravitational forces acting on the droplet. For the purposes of modeling the movement of coalesced droplets in fibrous filters, the force required to overcome the static force holding the droplet in place needs to be known. A number of previous studies of droplets on fibers have dealt with the stability of droplet conformations, contact angles, and droplet geometry.10-17 Some of this work details the measurement of the contact angle of a droplet on a cylindrical fiber10,14,18
)
The dynamics of wetting has received significant attention1,2 and remains of interest today.3 However, previous work has predominantly considered droplets on flat surfaces, with comparatively little work on cylindrical elements (such as fibers). Axisymmetric (or barrel-shaped) droplets (defined by contact angles ,90°) on cylindrical elements exist in both nature and industry. Understanding the behavior and properties of such droplets is important to the fields of surface chemistry and interface science. In addition, understanding the parameters governing the motion of these droplets is important to textile manufacturing, surface coating applications, and gas-liquid separation. The motion of these coalesced droplets in fibrous gas/oil separation filters is the motivation for this work, though there are a number of other areas where this work is important, such as the collection/removal of water from proton exchange membrane (PEM) fuel cells, where fibrous media analogous to aerosol filters are employed. Fibrous filters are commonly used to separate liquid aerosols from gas streams. However, despite the prevalence of such technology in industry, relatively little is known about the behavior of these systems, particularly on the microscopic level. Currently available models are of a (semi)empirical4,5 nature and as such are applicable only over a narrow set of parameters (fiber diameter, packing density (solidity), and oil type to name a few). These models also typically aim to predict macroscopic properties such as pressure drop4,5 and saturation.4 The exception here is Current address: School of Environmental Engineering, Griffith University, Nathan QLD 4111, Australia *To whom correspondence should be addressed. E-mail: r.mead-hunter@ curtin.edu.au. (1) de Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827–863. (2) Semal, S.; Blake, T.; Geskin, V.; de Ruijter, M.; Castelein, G.; De Connick, J. Langmuir 1999, 15, 8756–8770. (3) Seveno, D.; Valiant, A.; Rioboo, R.; Adao, H.; Conti, J.; De Connick, J. Langmuir 2009, 25, 13034–13044. (4) Raynor, P. C.; Leith, D. J. Aerosol Sci. 2000, 31, 19–34. (5) Frising, T.; Thomas, D.; Bemer, D.; Contal, P. Chem. Eng. Sci. 2005, 60, 2751–2762. (6) Hinds, W. C. Aerosol Technology: Properties, Behavior, and Measurement of Airborne Partices, 2nd ed.; Wiley: New York, 1999.
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(7) Lee, K. W.; Liu, B. Y. H. Aerosol Sci. Technol. 1981, 1, 35–46. (8) Lee, K. W.; Liu, B. Y. H. Aerosol Sci. Technol. 1982, 1, 147–161. (9) Mullins, B. J.; Kasper, G. Chem. Eng. Sci. 2006, 61, 6223–6227. (10) Bauer, C.; Dietrich, S. Phys. Rev. E 2000, 62, 2428–2438. (11) Brochard, F. J. Chem. Phys. 1986, 84, 4664–4672. (12) Carroll, B. J. Langmuir 1986, 2, 248–250. (13) Carroll, B. J. J. Colloid Interface Sci. 1984, 97, 195–200. (14) Carroll, B. J. J. Colloid Interface Sci. 1976, 57, 488–495. (15) McHale, G.; Newton, M. I. Colloids Surf., A 2002, 206, 79–86. (16) McHale, G.; K€ab, N. A.; Newton, M. I.; Rowan, S. M. J. Colloid Interface Sci. 1997, 186, 453–461. (17) Quere, D.; Di Meglio, J.-M.; Brochard-Wyart, F. Rev. Phys. Appl. 1988, 23, 1023–1030. (18) Wagner, H. J. Appl. Phys. 1990, 67, 1352–1355.
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image, without the need to measure L. Using the equation for the profile of a droplet,14,20 the relationship between the inflection angle (θi) (i.e., the angle between the fiber and the point on the droplet that has maximum slope) and the contact angle can be established,21 where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ai ¼ nð1 þ 2 tan2 θi Þ ( n ð1 þ 2 tan2 θi Þ2 - 1 ð5Þ and θ ¼
Figure 1. Scanning electron microscope (SEM) image of a fibrous filter.
(which is different from the contact angle that the same liquid would form on a flat plate1,17,19) and provides equations describing the profile of a droplet.14,20 Using the equations of Carroll,14 the reduced droplet volume (Vh), and the volume of the droplet (V), the wetted area of the fiber and the wetted length (L) (distance between contact lines) can be determined numerically from the known contact angle (θ), fiber radius (h), and droplet radius (measured perpendicular to the fiber (x2)). The reduced volume can be defined as V 2πn ð2a2 þ 3an þ 2n2 ÞEðφ, kÞ - a2 Fðφ, kÞ V ¼ 3 ¼ h 3 1 ð1Þ þ ðn2 - 1Þ1=2 ð1 - a2 Þ1=2 - πL n where E(φ, k) and F(φ, k) are elliptical integrals of the first and second kind, respectively, and Lh is the reduced wetted length, L ¼ 2½aFðφ, kÞ þ nEðφ, kÞ
ð2Þ
with the parameter a (used to simplify the rather complicated expression for the profile of a droplet14) defined as a ¼
x2 cos θ - h x2 - h cos θ
ð3Þ
and the reduced radius (n) defined as n ¼
x2 h
ð4Þ
Using these equations, however, relies on the contact angle being known or measurable because eq 3 depends on the value of θ. Direct measurement of the contact angle from a fiber image is not always reliable because there is a high degree of curvature at the fiber-liquid interface.21 The contact angle can be calculated from the Carroll14 equations if the wetted length, L, is known; however, this may be difficult to measure accurately because the exact boundary of the droplet is difficult to detect optically. However, the inflection method of Rebouillat et al.21 allows the contact angle of a droplet on a fiber to be determined from an (19) Bacri, J.; Frenois, C.; Perzynshi, R.; Salin, D. Rev. Phys. Appl. 1988, 23, 1017–1022. (20) Yamaki, J.-I.; Katayama, Y. J. Appl. Polym. Sci. 1975, 19, 2897–2909. (21) Rebouillat, S.; Letellier, B.; Steffenino, B. Int. J. Adhes. Adhes. 1999, 19, 303–314.
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qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 - n2 Þða2i - 1Þ nai þ 1
ð6Þ
with ai < 1. Here ai is a modified value of a used to derive the relationship between θ and θi. It is therefore possible to measure all necessary parameters for the equations from an image of the droplet, which allows the comparison of force measurements of different droplets to a chosen droplet parameter. The atomic force microscope (AFM) is a very versatile apparatus and can be used to measure forces accurately if both the spring constant of the cantilever (ks) and the signal sensitivity (ss) of the detector are known. The force (Fm) required to move a coalesced droplet can be measured using the atomic force microscope (AFM) by employing the lateral force microscopy technique. Fm can be determined from the measured cantilever deflection (Dc) using Fm ¼
ks Dc ss
ð7Þ
There are a number of methods available22-27 that allow these parameters to be determined. There has been only one previous study of coalesced droplets on fibers using the AFM;28 however, in this work the force required to pull the droplet off the fiber was measured rather than the force used to move the droplet along the fiber. This work will attempt to determine the forces required to slide a droplet along a fiber both theoretically and experimentally using the AFM.
Model For the purposes of modeling the force required to move a droplet from its static position on a fiber, we shall consider an axisymmetrical droplet situated on a cylinder of radius h, oriented horizontally, as shown in Figure 2. The center of mass, G, of the coalesced droplet may be displaced a distance r from the center of the fiber. (For a droplet at its axisymmetrical rest position, where the effects of gravity are negligible, r = 0.) Here, b is the radius of the droplet defined as the distance from G to the contact line in contrast to the value of x2, which is the perpendicular distance from the center of the fiber to the droplet edge. (These values are different because the droplets are not perfectly spherical.) In the case where r = 0, b is half the wetted length (the distance between (22) Palacio, M.; Bhushan, B. Crit. Rev. Solid State Mater. Sci. 2010, 35, 73–104. (23) Cannara, R.; Eglin, M.; Carpick, R. Rev. Sci. Instrum. 2006, 77, 053701-1– 053701-11. (24) Green, C.; Lioe, H.; Cleveland, J.; Proksch, R.; Mulvaney, P.; Sader, J. Rev. Sci. Instrum. 2004, 75, 1988–1996. (25) Sader, J.; Chon, J.; Mulvaney, P. Rev. Sci. Instrum. 1999, 70, 3967–3969. (26) Sader, J. Rev. Sci. Instrum. 1995, 66, 3789–3798. (27) Cleveland, J. P.; Manne, S.; Bocek, D.; Hansma, P. K. Rev. Sci. Instrum. 1993, 64, 403–405. (28) Mullins, B. J.; Pfrang, A.; Braddock, R. D.; Schimmel, T.; Kasper, G. J. Colloid Interface Sci. 2007, 312, 333–340.
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is met, then W ¼
2xA 2 þ 2yA 2 - 2ryA zA
ð18Þ
Vector v at A acts in the direction of the surface tension at A and is normalized to provide a unit vector vˆ for each value of ψ and therefore for all points around the contact line. This normalized vector is then 2ixA þ 2jðyA - rÞ þ kWÞ v^ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4x2A þ 4ðyA - rÞ2 þ W 2
Figure 2. Schematic representation of the model parameters with the droplet (a) at the axisymmetrical rest position and (b) displaced from this position by a distance r. The x axis is perpendicular to the yz plane, and the origin is at the center of mass, G.
The force required to move the droplet from its static (stable) position will then be a function of vˆ and the advancing (τa) and receding (τr) line tensions such that Z 2π Z 2π Fm ¼ τa v^ dψ þ τr v^ dψ ð20Þ 0
ð8Þ
yA ¼ r þ h cos ψ
ð9Þ
zA 2 ¼ b2 - r2 - h2 - 2rh cos ψ
ð10Þ
where ψ denotes the angular position on the contact line around the fiber, so 0 e ψ e 2π. If we consider the position vector (rA) of a point A on the contact line, we get pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rA ¼ ih sin ψ þ jðr þ h cos ψÞ þ k b2 - r2 - h2 - 2rh cos ψ ð11Þ where i, j, and k are the component vectors. The plane tangent to the droplet at point A is given by r 3 uA ¼ b
ð12Þ
where r is the position vector of a point in the tangent plane and r uA ¼ A ð13Þ jrA j is a unit vector normal to the droplet at A. The normal to the fiber uf at A is given by uf ¼ 2ðxA Þi þ 2ðyA - rÞj þ Wk
ð14Þ
The tangent plane, perpendicular to the contact line at A, may be displaced by vs such that r ¼ rA þ vs
ð15Þ
where s is a parameter and the vector v has the form v ¼ 2ixA þ 2jðyA - rÞ þ kW
ð16Þ
If the condition v 3 uA ¼ ð2ixA þ 2jðyA - rÞ þ kWÞ 3 Langmuir 2011, 27(1), 227–232
rA ¼ 0 jrA j
ð17Þ
0
where zA > 0 for the advancing contact lines and zA < 0 for the receding contact lines and must be real. The line tensions are
contact lines). The position of a point on the contact line around the fiber can then be expressed in terms of xA ¼ h sin ψ
ð19Þ
τa ¼ 2hπσ cos θa
ð21Þ
τr ¼ 2hπσ cos θr
ð22Þ
for the advancing and
for the receding boundaries, with σ being the surface tension of the liquid and θa and θr being the advancing and receding contact angles, respectively.
Experimental Section This work required the preparation of modified cantilevers because the contact between the AFM probe and the droplet needed to be sufficiently large that the droplet could be moved along the fiber with the AFM probe, and it was found that the surface of the AFM probe needed to be rendered oleophobic to ensure that the droplet remained on the fiber and did not attach to the cantilever. These two criteria were achieved by mounting a colloidal probe with a surface coating onto a cantilever. A glass sphere was attached to the cantilever (NP-OW, tipless silicon nitride cantilevers, Veeco, Camarillo, CA) using an epoxy resin (Araldite, Selleys, Australia). The surface of the attached sphere was then coated with a fluorinated copolymer (TL1143, Thor Chemie, Speyer, Germany), rendering it oleophobic. The size of the coalesced droplet that could be moved by a cantilever was found to depend on the size of the glass sphere attached to the cantilever. For this work, a series of cantilevers were prepared using glass spheres ranging from 20 to 70 μm in diameter. The fibers (9.1-, 10.2-, 12.5-, 17-, and 22.7-μm-diameter polyester, Barnet, Aachen, Germany) were mounted individually in a custom-made stainless steel mount (SS 304) so that a length of approximately 50 mm of fiber was held rigidly 5 mm above the base of the mount. The fibers were then loaded (with oil) by exposing them to oil aerosol generated by a collision nebulizer. Hence, the fibers were allowed to collect the oil in the same manner that they would in a mist filter. The size of the coalesced droplets that accrete on the fiber was dependent on the time that the fiber was exposed to the aerosol stream. The oils used in this work were (A) a monocomponent laboratory oil, diethyl-hexylsebacate (DEHS, Sigma-Aldrich), and (B) a multicomponent commercial lubricating oil, RX Super (Castrol). The properties of these oils are given in Table 1. The contact angles of these oils on the fiber were determined using the inflection method of Rebouillat et al.21 from images taken with a CCD camera. The droplet dimensions were determined DOI: 10.1021/la104147s
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density (F) kg/m3 viscosity ( μ) Pa 3 s surface tension (σ) N/m
DEHS (A)
RX super (B)
914 0.025 0.0324
875 0.131 0.0295
Figure 4. Static contact angles for each oil and fiber combination used in this work. Error bars indicating an average standard deviation of 1.5° are omitted for clarity.
Figure 3. Coalesced droplet of oil B being measured on a 22.7-μmdiameter fiber. optically and by using the Carroll14 equations evaluated in MATLAB (MathWorks). For the AFM measurements, a Digital Instruments Dimension 3100 scanning probe microscope system with Nanoscope V6.13 control software (Veeco) was used. All force measurements were made in contact mode, whereas the topography of the fiber surface was imaged in tapping mode. The torsional spring constant of the cantilevers used was determined via the torsional Sader method,24 and the lateral signal sensitivity was measured according to the method of Cannara et al.23 The choice to use the torsional Sader method24 to determine the torsional spring constant was made because it uses parameters that can be measured either optically or using the tuning function in the AFM. However, the conversion of the spring constant to a lateral stiffness does require the cantilever thickness to be known, so for this, the nominal values supplied by the manufacturer were used. The errors associated with the lateral force calibration using these methods22-24 are approximately (15%. To move the droplets along the fiber and to measure the required force, the probe was moved at a scan angle of 90° with the slow scan axis disabled. Figure 3 shows the typical setup for the measurements.
Results and Discussion For a detailed analysis of the force measurements, information on the dimensions of the measured droplets is required. The first step was to determine the contact angle of the coalesced droplets on each of the different fiber diameters. Figure 4 shows the contact angles for all of the combinations examined in this work. The contact angles of axisymmetrical droplets of both oils were measured for each fiber diameter. A minimum of 20 droplets were measured for each oil-fiber combination, which were then averaged to give a representative static contact angle for each of the fiber diameters with each oil (Figure 4). With respect to our model, dynamic contact angles would be preferred. However, the measurement of the dynamic contact angle is exceedingly difficult, and for this work, the static contact angles have been used. 230 DOI: 10.1021/la104147s
Figure 4 shows that there is little difference in contact angles for the two oils on each of the different fiber diameters, with the exception of the thinnest fibers where the difference becomes evident. Given the similar surface tensions of the oils, similar contact angles were expected, with the deviation at the small fiber diameters perhaps being due to liquid surface tension forces having a greater influence as the fiber diameter is decreased. It was found that the force required to move a droplet on a fiber increased with increasing tip velocity, which is likely due to there being both a static and a dynamic component of the measured force. The dynamic (viscous drag) force component is dependent on the tip velocity (ut) and therefore is a result of measuring the force required to move a droplet by actually moving it. The static component of the force (due to static friction/interfacial tension), though, is the force holding the droplet in place and as such is present when the droplet is not in motion (i.e., when ut = 0). By using velocities from as low as 0.8 μm/s and up to 50 μm/s, a series of measurements were made for each droplet so that the relationship between tip velocity and cantilever deflection could be found. The deflection corresponding to the movement of the droplet was extracted from each of the scans at the different tip velocities and converted to a force using eq 7, resulting in a plot of force versus tip velocity as shown in Figure 5. It should be mentioned that each of these points represents the force required to move the droplet, so elastic tension and the resultant droplet deformation have already occurred before the droplet moves and the force measurement is obtained. An examination of Figure 5 reveals that there is an increase in the force with tip velocity, which can be described by a polynomial fit. For the case shown in Figure 5 the total force, Ft, is described by the polynomial fit Ft ¼ - 0:0144ut 2 þ 4:4962ut þ 40:605
ð23Þ
where ut is the tip velocity and the fit has a correlation coefficient (R2) of 0.99. Remembering that the force here consists of both a static and a dynamic component, eq 23 can be evaluated as the value of ut approaches 0 such that Ft approaches the force holding the droplet in place when there are no dynamic forces acting. For the droplet measured in Figure 5, it is 40.6 μN. This is the static/ minimum force required to commence droplet motion. The model was evaluated for droplets of the same dimensions as those measured using eqs 1 and 2. The droplet parameters were determined using the value of x2 measured directly from the droplet images and the contact angle data from Figure 4 corresponding to the oil type and fiber diameter being used. For the Langmuir 2011, 27(1), 227–232
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Figure 5. Force vs tip velocity curve used for the static force determination. This is for a droplet on a 17 μm polyester fiber.
Figure 7. Measured and modeled static forces for coalesced droplets of oil B on 9.1-, 10.2-, 12.7-, 17-, and 22.7-μm-diameter fibers.
Figure 8. Topography of a section of the surface of a 17-μmdiameter polyester fiber.
Figure 6. Measured and modeled static forces for coalesced droplets of oil A on 9.1-, 10.2-, 12.5-, and 17-μm-diameter fibers.
purpose of modeling the droplets, the effect of gravity is considered to be negligible and r = 0 for all droplets. The model neglects the surface energy of the fiber because this affects the wettability and as such is intrinsic to the contact angle (which has been measured). Figures 6 and 7 show the results plotted in terms of force over wetted length versus reduced droplet volume (as defined by eq 1). This allows all the results for the different fiber diameters to be plotted on the same set of axes. For oil A, the correlation coefficient (R2) between measured and modeled values is 0.77, and for oil B, it is 0.82. A close examination of the modeled values revealed that an empirical fit could be made to the model data points. The best empirical fits were found using power laws. These are Fl ¼ 6:8012V
- 0:241
ð24Þ
Fl ¼ 7:9946V
- 0:282
ð25Þ
for oil A and
for oil B, where Fl is the force per wetted length (Fm/L) and the correlation (R2) between the modeled data and the power law fit is 0.99 for both oils. Although we are able to determine all the values necessary for the model in eq 20, the empirical fits allow a simplified force determination without the need for a numerical Langmuir 2011, 27(1), 227–232
evaluation of the model. It is believed that these equations may be useful in fibrous filter (and other engineering) applications. Therefore, it is possible to determine the force required to move a coalesced droplet, of known dimensions, of oil A or oil B from its stationary position without the need to evaluate the force model numerically or to evaluate the model for a reduced number of points. Figures 6 and 7 both show some scattering of data points in terms of the modeled values and the model fit. This is not unexpected because we are dealing with droplets of different wetted lengths on different diameter fibers with contact angles that vary over the different fiber diameters. It is also possible that measurement errors associated with the determination of the droplet dimensions may affect the model. The variation between modeled and measured forces for both oils A and oil B are due to a number of factors. The error associated with the force calibration of the cantilevers must of course be considered, but there are also two possible explanations for this based on experimental observations and measurements. First, the fibers are not smooth cylinders; they are polymer fibers manufactured using the melt-blown process and contain surface inhomogeneities that will affect the length and shape of the contact line, wetted area, and ultimately the force required to move the droplet. Figure 8 shows the 3D tapping-mode AFM topography image of a randomly selected fiber section, where the surface irregularities can clearly be seen. Second, it was observed that during some measurements the droplet appeared to move into contact with the sphere on the cantilever and the cantilever was still approaching the fiber. This leads to a situation where the droplet is no longer in the axisymmetrical DOI: 10.1021/la104147s
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Figure 9. Coalesced droplets on a fiber subject to air flow.29 The displacement from the preferred axisymmetrical position is evident.
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The effect of the droplets shifting away from their axisymmetrical rest position was investigated by deliberately pulling the droplet to the side of the fiber with the cantilever and measuring the force required to move the droplet along the fiber from this new position. This was also modeled by varying r in the model. A comparison of the experimental and modeled data is shown in Figure 10. Figure 10 shows a gradual deviation between the measured and modeled values with increasing r. This is likely because the contact angle may vary at different points around the fiber; for example, when r is increased, the symmetry of the droplet around the fiber will be lost. However, both modeled and measured values do demonstrate that as the value of r increases the force required to move the droplet from its static position decreases. This may have practical applications in mist filtration because it implies that the onset of droplet drainage (motion) could be brought forward by increasing the flow velocity through the filter and moving the coalesced droplets away from their stable (rest) positions.
Conclusions
Figure 10. Influence of increasing droplet displacement (r) on force (Fm).
rest position (i.e., r 6¼ 0) and the force measured is no longer the force required to move the droplet from its original (static) position but from a slightly off-center static position. This is important both in terms of explaining differences between measured and modeled values and also in terms of the behavior of mist filters. It has been observed in fibrous filter systems that under the influence of air flow a droplet may be pushed away from its preferred position (Figure 9). (29) Mullins, B. J.; Braddock, R. D.; Agranovski, I. E.; Cropp, R. A. J. Colloid Interface Sci. 2006, 300, 704–712.
232 DOI: 10.1021/la104147s
The force required to move a coalesced liquid droplet along a fiber from its stable stationary position has been measured. We have found good agreement between the measured force values and those predicted by our model and have identified factors that may influence the static force. These include fiber surface inhomogeneity and displacement of the droplet from the axisymmetric rest position. The influence of pulling the droplet away from the fiber on the force required to move it along the fiber has also been established. It is expected that the theoretical model presented would show very close agreement with measured data if atomically smooth fibers could be obtained and AFM measurement error was neglected. Additionally, empirically derived equations have been given for the specific cases examined, which allow the calculation of the force required to move a droplet on a fiber and are based on the reduced volume, without the need to solve the theoretical model numerically. The theoretical model requires h, b, θ, and σ as input to calculate the force. As shown, the model conforms to a power law relationship for the cases evaluated in this work. This implies that for a new case the model can be evaluated for a relatively small number of points and the power law can be fitted to obtain a range of values. Acknowledgment. This work was supported by an Australian Research Council Linkage Grant (LP0883877).
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