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Attractive Forces between Surfaces: What Can and Cannot Be Learned from a Jump-In Study with the Surface Forces Apparatus? Olga I. Vinogradova*,†,‡ and Roger G. Horn§ Max-Planck-Institute for Polymer Research, Postfach 3148, 55021 Mainz, Germany, Laboratory of Physical Chemistry of Modified Surfaces, Institute of Physical Chemistry, Russian Academy of Sciences, 31 Leninsky Prospect, 117915 Moscow, Russia, and Ian Wark Research Institute, University of South Australia, Mawson Lakes, South Australia 5095, Australia Received November 2, 2000. In Final Form: December 21, 2000 We study theoretically the dynamics of film thinning under the action of an attractive surface force near the point of a jump instability. Our approach is illustrated by modeling van der Waals and hydrophobic attractive forces. The main result is that with the hydrophobic force law reported previously it is often impossible to establish the jump separation with any certainty. The surfaces instead approach slowly from a distance which is much larger than the point where an actual jump is expected. We conclude that an attractive force measured by the static jump technique is overestimated, and we formulate principles of a new dynamic jump method. The use of this new technique would permit direct measurements of attractive forces at separations below the static jump distance down to contact of the surfaces.
I. Introduction Measurements of attractive forces are important for understanding various phenomena. For example, hydrophobization of solids plays a crucial role in adhesion, cavitation, dewetting, film stability, and coagulation, and this can be related to the long-range attractive forces1 acting between hydrophobic surfaces. Several different instruments have been used for measuring forces between hydrophobic surfaces in water. One significant experimental technique is the surface force apparatus (SFA). With an SFA attractive forces can be measured in different ways. In the original version2 the surface force Fs(h) (negative for attraction) is simply equated to the restoring force of the cantilever spring Fk on which one of the surfaces is mounted. This condition for equilibrium is equivalent to the first distance derivative of the total energy of the system being zero. In order for the equilibrium to be stable, the second derivative should be positive, which gives k > dFs/dh, where k is the stiffness of the cantilever spring. When the condition for stability is violated, the surface separation h jumps from one value to another, making certain portions of the force-distance curve inaccessible to measurement. For this reason the static deflection technique is not very suitable for measurements of a strongly attractive interaction because the range of measurement is limited. Increasing the spring constant will increase the range of stability, but only at the expense of sensitivity in measuring the force. There are some measurements of hydrophobic forces using the * To whom correspondence should be addressed. E-mail:
[email protected]. † Max-Planck-Institute for Polymer Research. ‡ Russian Academy of Sciences. § University of South Australia. (1) Christenson, H. K. In Modern approaches to wettability: Theory and applications; Schrader M. E., Loeb, G., Eds.; Plenum Press: New York, 1992. (2) Tabor, D.; Winterton, R. H. S. Proc. R. Soc. London A 1969, 312, 435.
static deflection method,7-10 but the more common way to obtain information on attractive forces is the static jump method, in which the onset of the unstable regime is sought for different values of the spring constant,3 i.e., those points where dFs/dh ) k. Another possibility that has been used on a few occasions is a drainage technique,4 in which the surface force Fs is found by subtracting the hydrodynamic force Fh from the total force measured.5,6,11,12 Hydrophobic (and other) force measurements are performed in water solutions, i.e., in a viscous liquid. One can therefore suggest that when using the static jump method, the loss of mechanical stability of the system could appear not as an abrupt jump but as a time-dependent film drainage. The aim of this paper is to study numerically the dynamics of film thinning near the point of a jump instability caused by an attractive surface force. The idea of studying the dynamics of a jump was earlier proposed for a situation in which there are negligible viscous forces in the system.13 This investigation and another more recent one14 were also concerned with speeds that are much higher than those typically encountered in an SFA, and it focused on the inertial term in the equation of motion. Here we study the damping term, which is far more important for the liquid systems. (3) Israelachvili, J. N.; Adams, G. E. J. Chem. Soc., Faraday Trans. 1 1978, 74, 975. (4) Chan, D. Y. C.; Horn, R. G. J. Chem. Phys. 1985, 83, 5311. (5) Claesson, P. M.; Christenson, H. K. J. Phys. Chem. 1988, 92, 1650. (6) Vinogradova, O. I. Langmuir 1998, 14, 2827. (7) Israelachvili, J. N.; Pashley, R. M. Nature 1982, 300, 341. (8) Pashley, R. M.; McGuiggan, P. M.; Ninham, B. W.; Evans, D. F. Science 1985, 229, 1088. (9) Hato, M. J. Phys. Chem. 1996, 100, 18530. (10) Kurihara, K.; Kunitake, T. J. Am. Chem. Soc. 1992, 114, 10927. (11) Christenson, H. K.; Fang, J.; Ninham, B. W.; Parker, J. L. J. Phys. Chem. 1990, 94, 8004. (12) Christenson, H. K.; Claesson, P. M.; Berg, J.; Herder, P. C. J. Phys. Chem. 1989, 93, 1472. (13) Lodge, K. B.; Mason, R. Proc. R. Soc. London A 1982, 383, 279, 295. (14) Attard, P.; Schulz, J. C.; Rutland, M. W. Rev. Sci. Instrum. 1998, 69, 3852.
10.1021/la001534g CCC: $20.00 © 2001 American Chemical Society Published on Web 02/03/2001
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II. Analysis We use the following physical model of the SFA which measures the force between crossed cylindrical surfaces. The lower solid is fixed to a cantilever spring. The upper solid is mounted rigidly and may be moved to give variations in the separation h(t) between the surfaces. The drive function L(t) represents the distance between surfaces in the absence of any forces between them (no spring deflection), so that the instantaneous deflection ∆(t) of the spring from its equilibrium position is given by ∆(t) ) h(t) - L(t). The separation between the two cylinder surfaces as a function of time is calculated by solving the equation of motion of the lower solid, under the assumption (justified below) that its inertia is negligible. Then the total force, which is the sum of hydrodynamic and surface forces, is balanced by the restoring force of the cantilever spring: Fs + Fh ) Fk, with Fk ) k∆(t). In the case of a deflection or a jump experiment, the movable surface is driven toward the other surface in a series of steps. Because each step in L occurs over a time that is short compared to the time required for the surfaces to reach their new equilibrium separation, there is negligible error in modeling the step as instantaneous. Hence, L(t) ) h0 + s + vdt - ∆0, where h0 is the initial separation, ∆0 is the initial deflection, and s < 0 is the size of a drive step. This expression also allows for the possibility of drift at a speed vd, which is negative when the surfaces drift toward each other. This arises because of differential expansion of components of the apparatus. For the SFA temperature changes of order 10-2 K move the surfaces by 1 nm, and drift speeds of order 10-2 nm/s are commonly encountered. With this drive function, the equation to be solved is
Fs + Fh ) k(h - h0 - s - vdt + ∆0)
(2.1)
We model k in the interval from 8 × 104 to 5 × 105 mN/m and s from -5 down to -0.1 nm. For thin Newtonian films having viscosity η and assuming slip boundary conditions at the water-hydrophobic solid interface,15,16 this is readily calculated by exploiting the lubrication approximation, which, for cylinders having the same radius R and the same slip length b, leads to17,18
Fh dh h 6b 2πηR )(2.2) 1+ ln 1 + -1 R b 6b h dt
[(
) (
) ]
We set R ) 1 cm and use slip lengths of 0, 10, and 100 nm, which, we are confident, would encompass realistic situations. This analysis only includes attractive forces, which is the situation preferred experimentally to avoid having to subtract out a double layer force.5,19,20 The attractive force is the sum of a nonretarded van der Waals attraction and a hydrophobic force. The hydrophobic attraction inferred from an SFA experiment can be modeled with a doubleexponential decay:1
( )
( )
Fs A h h ) - 2 - B1 exp - B2 exp (2.3) R λ λ 6h 1 2 (15) Vinogradova, O. I. Int. J. Mineral Proc. 1999, 56, 31. (16) Barrat, J. L.; Bocquet, L. Phys. Rev. Lett. 1999, 82, 4671. (17) Vinogradova, O. I. Langmuir 1995, 11, 2213. (18) Vinogradova, O. I. Langmuir 1996, 12, 5963. (19) Christenson, H. K.; Claesson, P. M. Science 1988, 92, 1650. (20) Kekicheff, P.; Spalla, O. Phys. Rev. Lett. 1995, 75, 1851.
where A ) 2.2 × 10-20 J is the Hamaker constant for mica across water,3 B1 and B2 are the prefactors, and λ1 and λ2 are the decay lengths. The long-range hydrophobic component was modeled with B1 ) 0-2 mN/m and λ1 ) 10 nm and the short-range with B2 ) 0-5 mN/m and λ2 ) 1 nm. Differential equation (2.1) together with eqs 2.2 and 2.3 is solved numerically to give h(t). First, we set h0 and specify the steps s in L and the time intervals between steps. Before the first step, we calculate ∆0 from the force acting at h0 (including the possibility of a hydrodynamic component due to drift). To simulate a typical experiment, we then run the program for a certain time tf (usually 30-100 s), which is intended to be long enough for decay down into the equilibrium state (or into the final state at tf in the case of vd * 0). At the end of this time, we reset the clock to t ) 0, make an abrupt step in the drive, and follow the dynamic behavior again for a further time tf. To make every subsequent step, we reset the clock to t ) 0 and set h0 ) h and ∆0 ) ∆ at the end of the previous step. The procedure is continued until the liquid film drains to h ) 0. III. Results and Discussion Figure 1A shows a set of typical separation versus time curves expected in the no-slip limit with a single (longrange) exponential function as a model for the hydrophobic force. At separations much larger than the static jump point, all of the zero-drift curves decay down into the final equilibrium state after a step in the drive function is made. Although successive steps all have the same size in the drive L, the steps in h increase as the static jump point is approached. In addition, the relaxation time increases for successive steps,21 because of a decrease in h and an increase in ∆. When h0 approaches the separation of a jump and the corresponding equilibrium separation h is less than the jump distance, the following behavior is seen. At first the surfaces move quite slowly together, appearing to approach an equilibrium state. In some situations a jump could easily be implicated by or even confused with thermal drift. After the static jump separation is reached the surfaces begin to approach faster, gradually accelerating until they reach contact. However, in the typical experimental situation, |dh/dt| would never exceed 100 nm/s. This means that the Stokes number, which provides a measure of the solid’s inertia relative to the viscous forces and is defined as St ) m |dh/dt|/ 6πηR2,22,23 where m is the mass of a spring-mounted surface, never exceeds ∼5 × 10-5 , 1. Hence, our system is strongly overdamped and has negligible inertia. We remark that such a speed is low enough to be recorded on a normal video system.4,5 In general, depending on the drift speed, times ranging from 18 to 130 s are required for surfaces to move into contact from the last stable position. It is important to note that the last stable position depends on the history, i.e., on the initial position, on the previous steps, and on the thermal drift. Hence, the last stable position h0 cannot readily be controlled, it can occur at separations significantly larger than the instability point, and the accuracy of measuring the static jump distance cannot be improved dramatically by taking smaller drive steps. Thus, for the parameters used in (21) Horn, R. G.; Hirz, S. J.; Hadziioannou, G.; Frank, C. W.; Catala, J. M. J. Chem. Phys. 1989, 90, 6767. (22) Davis, R. H.; Serayssol, J. M.; Hinch, E. J. J. Fluid Mech. 1986, 163, 479. (23) Vinogradova, O. I.; Feuillebois, F. J. Colloid Interface Sci. 2000, 221, 1.
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Figure 1. Typical sequence of separation vs time curves expected with k ) 102 N/m, tf ) 40 s, s ) -1 nm, B1 ) 0.5 mN/m, B2 ) 0 mN/m, and h0 ) 33 nm. After each 40 s interval, the next segment of the curve restarts at t ) 0. The distance at which a static jump is expected is shown as a dashed line and is equal to 17.4 nm with the parameters used. (A) b ) 0, vd ) -0.01 (dotted), 0 (solid), and 0.01 (dash-dotted) nm/s. (B) vd ) 0; the dotted line represents the case b ) 100 nm, the dash-dotted line corresponds to b ) 10 nm, and the solid line plots the results for a no-slip limit.
Figure 1, the difference between the last stable position and the true instability point can be as much as 4.6 nm for a step size of -1 nm and 1.3 nm for s ) -0.1 nm. We recall that the usual claim for resolution in distance measurements and control in the SFA is 0.1 nm. It has recently been found14 that dynamic force measurements with an alternative surface force measurement device (MASIF) are seriously implicated by the inertia of solids (our estimates suggest that their St is well above 50; i.e., the system is indeed controlled by inertia), which leads to underestimation of the attractive force. Our results show that with the SFA static jump technique the attractive force can only be overestimated and not underestimated. [This analysis can also be applied to van der Waals forces (B1 ) B2 ) 0 in eq 2.3), and the same conclusion is reached. However, for the same experimental parameters, the uncertainty in establishing the true instability point (9.0 nm) is up to 2.9 nm for a step size of -1 nm and 0.80 nm for s ) -0.1 nm; i.e., the error
Vinogradova and Horn
Figure 2. Time derivative of the surface separation calculated with the same parameters as those in Figure 1 (in the absence of slip or drift). (A) Solid curves from right to left are the speeds calculated for the sequence of steps at ever-decreasing separations. The dashed line represents the static jump distance. (B) Curves from 1 to 7 correspond to the steps in the drive above the static jump distance. Curve 8 corresponds to a jump thinning.
would be less than that for a hydrophobic force. In contrast to hydrophobic forces which are generally predominant in the systems in which they are measured, van der Waals forces often appear in conjunction with a double-layer repulsion, which can significantly decrease the force gradient and make the determination of A much more accurate, or an oscillatory solvation force, whose presence would complicate the present analysis.] Slippage of water over the surfaces is responsible for a shorter relaxation time at large separations, as well as for the much faster approach below the point of a static jump (Figure 1B). The static jump point corresponds to a minimum in the hydrodynamic resistance force. Our calculations also suggest that in all cases, i.e., independently of the presence of a drift or slip, the separation of an actual jump is very close to a maximum in dh/dt (Figure 2), and therefore at this point the curvature of the drainage curve changes sign. This suggests a new possibility for studying the instability. Variation of the parameters in the governing equations (in particular k and B1) leads to a quantitative difference in the results, but it does not change the conclusions about
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the film thinning near a jump point. We note, however, that the addition of the second short-range exponent to the long-range one does not influence the behavior of the drainage curve near the point of the static jump, so that it cannot be responsible for an abrupt jump of surfaces if any. The observation of an abrupt jump could be a consequence of a strong (large prefactor) long-range force1 or serve as an indication of slippage,15 cavitation,19,24 or coalescence of surface-attached bubbles. In the last case, the abrupt jump is the result of a strong capillary force.25,26 In summary, the standard static jump technique often does not allow an accurate determination of the instability
point and leads to an overestimation of the long-range attractive force. Much more information could be obtained experimentally by studying the dynamics of a jump thinning as suggested here. The dynamic jump method opens up a new range of separations in which attractive forces can be measured down to contact and allows the static jump distance to be determined from the inflection point of the drainage curve. Investigation of the jump dynamics also permits a judgment about the possible origin of the attractive force that caused it and a study of a number of dynamic effects, such as, for example, slippage.
(24) Vinogradova, O. I.; Bunkin, N. F.; Churaev, N. V.; Kiseleva, O. A.; Lobeyev, A. V.; Ninham, B. W. J. Colloid Interface Sci. 1995, 173, 443. (25) Considine, R. F.; Hayes, R. A.; Horn, R. G. Langmuir 1999, 15, 1657. (26) Yakubov, G. E.; Butt, H. J.; Vinogradova, O. I. J. Phys. Chem. B 2000, 104, 3407.
Acknowledgment. This work was initiated while O.I.V. was an Alexander von Humboldt Fellow at the University of Mainz. C. A. Helm is thanked for comments on the manuscript. LA001534G
