Direct measurement of the viscoelectric effect in water.

The viscoelectric effect concerns the increase in viscosity of a polar liquid in an electric field due to its interaction with the dipolar molecules and was first determined for polar organic liquids more than 80 y ago. For the case of water, however, the most common polar liquid, direct measurement of the viscoelectric effect is challenging and has not to date been carried out, despite its importance in a wide range of electrokinetic and flow effects. In consequence, estimates of its magnitude for water vary by more than three orders of magnitude. Here, we measure the viscoelectric effect in water directly using a surface force balance by measuring the dynamic approach of two molecularly smooth surfaces with a controlled, uniform electric field between them across highly purified water. As the water is squeezed out of the gap between the approaching surfaces, viscous damping dominates the approach dynamics; this is modulated by the viscoelectric effect under the uniform transverse electric field across the water, enabling its magnitude to be directly determined as a function of the field. We measured a value for this magnitude, which differs by one and by two orders of magnitude, respectively, from its highest and lowest previously estimated values.

The viscoelectric effect concerns the change in the viscosity of polar liquids in the presence of an electric field (1–3). It arises from the interaction of the field with the dipolar molecules, and while its molecular origins are still not well understood (4–6), it has considerable relevance in areas ranging from surface potential measurements (7–9) and boundary lubrication (10) to nanofluidics and its applications (11–13). Knowing the magnitude of the viscoelectric effect is thus of clear importance. It was first measured by Andrade and Dodd (1–3) for a range of polar organic liquids, by monitoring their flow in a narrow channel between metal electrodes across which a known electric field was applied, and quantified via a viscoelectric coefficient f using an empirical relation based on their results:

a simplified analysis leading to such a relation is given in Ref. (8). Here, η0 is the unperturbed bulk liquid viscosity (i.e., in the absence of any field). For the case of water, however, the most ubiquitous and important polar liquid, measurement of its viscosity in the presence of a strong, uniform field presents a strong challenge (as discussed later in this section), and to our knowledge no such direct measurements have been reported. Over the past six decades, therefore, the magnitude of the viscoelectric effect in water has been only indirectly estimated by extrapolation from its values for organic liquids (8), from estimates of its effect on electrokinetic phenomena (11, 14–19), or by other approaches (7, 12, 20, 21). These estimated values, as expressed in the viscoelectric coefficient f, vary over more than three orders of magnitude, ranging from f ∼10−17–2.5 × 10−14 (V/m)2 (). For completeness, we note that results contradictory to the viscoelectric model have also been reported (22) (i.e., suggesting a decreased water viscosity in an electric field). The reasons for the large span of these estimated f values were attributed to various factors such as solid/liquid coupling, varying ionic sizes, and varying water permittivity (12, 19); however, while these factors may play some role, there is no evidence that they could lead to such large discrepancies.

We believe, rather, that the origin of the large variance in the estimated magnitude of the viscoelectric effect arises because none of the experimental studies on water to date in which the f values were estimated was direct, in the sense of probing how the water viscosity varied with field in a uniform electric field. In all cases, viscosity changes were assumed to occur only in the nonuniform, rapidly decaying electrostatic potential near charged surfaces immersed in water. Changes in electrophoretic mobility, electro-osmosis, or hydrodynamic dissipation or water mobility between similarly charged solid surfaces were then attributed to some mean viscosity increase in these thin surface-adjacent layers (7, 11, 12, 14–21). In practice, however, the effect on these electrokinetic phenomena of viscosity or water mobility changes in the thin layers where such nonuniform, rapidly decaying fields are present is not easy to quantify reliably, especially in the presence of salt ions (12). At the same time, measuring the viscosity of water in a uniform electric field between two surfaces at different potentials, as was done for the polar organic solvents (2, 3) and which would provide a direct determination of its viscoelectric effect, presents a considerable difficulty. This is due to two main factors and arises because, in contrast to organic solvents, water may self-dissociate. Firstly, the potential difference that may be applied between the surfaces across water is limited, if electrolysis is to be avoided (23, 24), and secondly, electrostatic screening implies that the field decays strongly (within a Debye screening length) away from the surfaces (25–27). Even in purified water with no added salt (as in the present study), the potential decays rapidly away from a charged surface (see, e.g., Fig. 1), so that to measure viscosity in a uniform field between two surfaces, one would require flow channels of width of order some tens of nanometers or less, presenting a major challenge.

Fig. 1.

Numerical solution to the nonlinearized PB equation with σmica = −8.1 mC/m2, ψgold = 0.07 V, and ion concentration cb = 8 × 10−5 M, corresponding to the conditions of Fig. 4. (A) Surface potential on the mica surface and surface charge on the gold surface as a function of separation D. (B) Average electric field approximated as (|ψgold − ψmica|/D). (C) Local potential ψ as a function of distance d from the mica surface for different separations D. Dashed line in larger-scale inset is an eye guide of a linear approximation.

Fig. 4.

(A)–(D) show representative dD/dt(D) profiles (additional profiles in ). The two pairs, A, B and C, D, were acquired, respectively, in two independent experiments at the same contact point for each experiment but at two different applied potentials onto the gold surface (see text). Each profile was obtained by averaging over three to four jump-in trajectories (see Materials and Methods). The black curves are the theoretical fits using the best f value determined directly from the plot in E, while for comparison, the predicted behavior corresponding to the extremal values f = 10−17 and 1.2 × 10−14 (m/V)2 estimated in the literature () are shown by the green and orange curves. (E) A plot of η/η0 versus E2 (see Eq. and text following equation) whose slope is f (Eq. , showing the best-fit value to be f = 9.9 ± 2.8 × 10−16 m2/V2. Error bars correspond to SE of mean (for η/η0 axis) and to estimated uncertainties in the PB fit parameters () for the E2 axis. (F) Comparing the theoretical surface force Ftot = FPB + FvdW with the force evaluated from the experimentally measured D(t), i.e., the right-hand side of Eq. , now using the directly determined f = 9.9 × 10−16 m2/V2 (E); the close fit over the entire D range contrasts with the discrepancy at D <∼40 nm in Fig. 3 where f was set to 0.

In the present study, we overcome this by directly probing the viscosity of purified water across which a uniform electric field acts while it is confined between two surfaces in a surface force balance (SFB). In our experiments, a molecularly smooth gold surface at a controlled (positive) surface potential approaches an atomically smooth mica surface at constant surface (negative) charge density, so that a known electric field acts across the water-filled gap of width D between them; moreover, this field is very close to uniform at the most relevant surface separations (D ≲ 30 nm, Fig. 1). The dynamics of approach is strongly modulated by the viscous damping due to squeeze-out of the water as D decreases, and hence by its viscosity in the uniform electric field; by monitoring the approach rate of the surfaces at high temporal (millisecond) and spatial (approximately angstrom) resolutions, we are able therefore to directly evaluate the magnitude of the viscoelectric effect (the value of f).

Results and Discussion

Electric Field Between Interacting Surfaces across Water.

The SFB has been traditionally used to measure interactions between two single-crystal, atomically smooth mica surfaces as a function of their separation D, where, in water, each mica surface is negatively charged because of the dissociation of potassium ions (25, 26, 28), corresponding to a constant-charge surface with charge density σmica. Detailed descriptions of the SFB appear elsewhere (28, 29). More recently, by replacing one of the mica sheets with a template-stripped, molecularly smooth gold surface (30) (rms roughness ∼2 Å, ; Fig. 2) and controlling the potential ψgold of the gold with a potentiostat in a three-electrode configuration (31, 32) (Fig. 2), it is possible to measure interactions between the surfaces at different potentials and thus with an electric field acting between them (32).

Fig. 2.

Schematics of the SFB setup (29) (Materials and Methods). (A) Schematic of the smooth gold surface coating a cylindrical lens, mounted on the piezoelectric tube (PZT), facing the back-silvered mica surface glued on a similar lens, in a crossed-cylinder configuration, supported by the spring k. (B) Images of FECO formed by white light interference between the gold and the silver layers were recorded at high frame rates (up to 1,000 fps) during the approach of the surfaces. Separation (D) was evaluated with subnanometer precision from the wavelengths of the fringe tips obtained from a double Gaussian fit to the pixel intensity profile (FECO image shown here taken at 100 fps for better contrast). (C) An electric potential (controlled by a potentiostat) is applied to the gold surface via a three-electrode configuration. The counter (C) and reference (R) electrodes were constructed with platinum wires. The working (W) electrode is the gold surface. ψapplied is imposed by the potentiostat.

Purified water with no added salt (Materials and Methods) was added to the SFB bath immersing the mica and gold surfaces. When a positive potential is applied to the gold, a long-ranged electrostatic attraction acts between the surfaces, as previously observed (28, 32). The dynamics of approach was initiated (at time t = 0, at a separation D = D0 > 300 nm, larger than the range of any surface interactions) by imposing a steady extension on the piezo crystal, on which the top (gold) surface is mounted, at a speed vapp < 10 nm/s. This drives the gold surface toward the lower mica surface, which is mounted on a spring of constant k. Onset of the long-range electrostatic attraction is observed at surface separations D ∼100 nm, and at lower D values, the surfaces jump into contact because of an Euler-like spring-instability (33). During the approach, the separation D(t) is obtained from the interference fringes of equal chromatic order (FECO, Fig. 2), using fast video recording (∼103 fps) at millisecond intervals. The typical noise in D(t) is ∼2 Å. The interaction between the surfaces as they approach arises from both conservative forces, i.e., electrostatic forces FPB described by the Poisson-Boltzmann (PB) equation and van der Waals (vdW) forces between the gold and mica surfaces across water, FvdW, together with hydrodynamic forces due to the finite approach speed of the surfaces. The relative motion of the surfaces during the approach obeys the following equation, as previously described (33):

The term on the left is the total conservative surface force Ftot = FvdW + FPB (), which is balanced by the inertial term (first term on right), the force due to distortion of the spring from its unperturbed state at separation D0 at t = 0 (second term on right), and the hydrodynamic resistance (third term on right), where R is the mean radius of curvature of the curved mica and gold surfaces and η is the water viscosity in the gap, which depends on the electric field through the viscoelectric effect (Eq. . The inertial term (where m ∼2 gm is the mass of the lower lens) is readily shown to be negligible compared to the other terms (33). In this study, the electroviscous effect (34–37), which describes the resistance to the flow of the counterions in the diffuse double layers adjacent to the gold and mica surfaces (and should not be confused with the viscoelectric effect, which is the subject of our study), is negligible because of the low ion concentration and may therefore be safely ignored in the hydrodynamic resistance term (). The electric field in the gap E(D) can be evaluated by the PB equation for given ψgold and σmica. ψgold is unknown a priori from the applied potential on the gold because of the usage of a pseudoelectrode (Fig. 2) but may be evaluated from the force versus distance profile F(D) as the surfaces approach (32); σmica is controlled by the equilibrium of surface cation adsorption, mostly dependent on the proton concentration in the immediate vicinity of the surface (26), and may likewise be determined from the F(D) profile (32).

Typical force versus separation approach profiles are shown in Fig. 3 (normalized as Ftot/R in the Derjaguin approximation), evaluated from the experimentally measured D(t) via Eq. on the assumption that the viscosity is at its unperturbed (i.e., zero field) bulk water value η0 throughout. This is a good approximation at separations D > 40 nm, where the mean electric field between the mica and gold surfaces is so low (see Fig. 1) that differences between η and η0 are negligible at all D > 40 nm. This allows the unknown parameters σmica, ψgold, and bulk concentration cb to be extracted through comparison with solutions of the PB equation (together with vdW forces) for D > 40 nm (details in Materials and Methods and ). The resulting theoretical fits are shown in Fig. 3 ; the deviation at D < ∼30 nm between the experimental curve deduced from Eq. and the theoretical force fit based on the PB and vdW equations is due entirely to the fact that the viscoelectric effect is not taken into account in Eq. , as confirmed in the next section. Inset in Fig. 3 is shown for comparison data from an earlier study (32) of interactions between mica and gold at the same applied potentials on the gold (0.2 V), in which σmica, ψgold, and bulk concentration cb were independently determined and were evaluated from Eq. ignoring the viscoelectric effect. These data show, as expected, close similarity to the present study at D > ∼40 nm (the slight differences at lower D values are likely due to the much lower video acquisition rate and thus the time resolution in the earlier work).

Fig. 3.

(A, B) Representative force versus surface separation profiles evaluated from the experimentally measured D(t) variation via Eq. , assuming unperturbed viscosity of water. The smooth curve is the predicted FBP + FvdW fitted to the region D > ∼40 nm and used to extract values of σmica, cb, and ψgold (see for details). The inset in B compares typical force profiles in our study (gray and red curves) with results from an earlier study by Tivony et al. (32) (blue dots) between mica and gold surfaces at the same applied gold potential (0.2 V), showing close correspondence for D > ∼40 nm.

Using these parameters obtained from the force profiles in Fig. 3, we may evaluate the potential on the mica surface ψmica, the charge density on the gold surface σgold, and the potential in the gap ψ(d) (0 < d < D) by numerically solving the nonlinearized PB equation () at varying separations D, as shown in Fig. 1.

Fig. 1 shows that as the separation D decreases, σgold approaches −σmica and ψmica approaches ψgold, as earlier discussed (28). Ultimately the two surfaces mutually neutralize when in contact. The mean electric field Ē = |ψgold − ψmica|/D is shown in Fig. 1. Although ψmica becomes less negative as D decreases, Ē increases because of the hyperbolically increasing factor 1/D and eventually plateaus as D approaches zero. For our analysis with the viscoelectric model (Eq. , E = Ē is an excellent approximation at small separations as ψ(d) becomes more linear in trend (Fig. 1) so that the field is essentially uniform for D < ∼40 nm. At large separations, our fits to Eq. are seen to be justified by the small magnitude of Ē; at an order of 0.004 V/nm or less, water viscosity is enhanced by ∼1.6% or less, assuming a viscoelectric coefficient of order 10−15 m2/V2 (see next section), which, as noted earlier, has negligible impact on the approach dynamics at D > 40 nm.

Magnitude of Viscoelectric Effect Determined from Surface Approach Dynamics.

To analyze the effect of the viscoelectric effect on the dynamics of approach, we evaluated dD/dt(D), Eq. —obtained by rearranging Eq. —from the D(t) variation determined directly from the video recording of the approach:

The damping term now manifests the viscoelectric effect in the enhanced value of η = η0(1 + f ⋅ E2). The advantage of using dD/dt(D) as the metric as opposed to F(D) is that, given that D(t) is the primary measurable through the FECO position (Fig. 3), the error propagated from D is minimized by allowing only one linear operation on the parameter.

Fig. 4 shows several typical dD/dt(D) profiles, taken at different extracted values of the surface parameters, all showing these common features: at large separations, dD/dt(D) is equal to the driving speed vapp of the piezo crystal (<10 nm/s), while at smaller separations, D < ∼80 nm, the approach velocity increases under the increasing electrostatic attraction while being modulated by the viscosity-dependent hydrodynamic stress as in Eq. . At D = Dmin ∼10 to 20 nm, a local minimum in dD/dt(D) is seen due to the decreasing driving term Ftot ⋅ D at small distances. Physically, this may be understood as follows: The electrostatic component of the attractive force Ftot (Eq. arises largely because of the expulsion of counterion pairs between the oppositely charged surfaces (38, 39), and its magnitude, for surface separations D of order the Debye screening length or less for our sphere-on-flat geometry, scales approximately as (1/D) (39). This is similar to the D variation of the damping term in Eq. , so that the surfaces initially accelerate toward each other under the increasing attractive force. For D < ∼30 nm in the conditions of our experiments, however, solution of the PB equation shows that the attraction varies more weakly than (1/D) (38), while the damping term continues to increase as (1/D), leading to the deceleration and eventual decrease in dD/dt (). The two pairs of figure panels, Fig. 4 show dD/dt(D) profiles measured in two independent experiments but at the same contact point in each experiment (i.e., identical values of cb and σmica for each pair of figures, different for each pair), with only the potential applied on the gold surface being varied. A higher applied potential on gold, from 0.07 V in Fig. 4 to 0.1 V in Fig. 4 and from 0.05 V in Fig. 4 to 0.08 V in Fig. 4D, results in higher jump-in speed at all distances as expected and ∼70 nm/s and 85 nm/s differences, respectively, at the dD/dt(D) local minima of Fig. 4 compared with Fig. 4 .

To evaluate the viscoelectric coefficient f, first we evaluated D(t) and dD/dt(D) with f = 0 from Eq. using the theoretically predicted Ftot with values of σmica, ψgold, and cb extracted directly from the large D (>40 nm) force profile as described previously. We then derivewherewhich can be evaluated for the range of E values in the most relevant distances 8 nm < D < 30 nm. A plot of η/η0 versus E2 is shown in Fig. 4, using values extracted from the dD/dt curves with least noise (Fig. 4 ) (for η/η0) and from the PB equation (for E2, Fig. 1). Data were binned at equally spaced intervals of E2 and averaged. From Eq. , the slope of this plot is simply the viscoelectric coefficient f, and its best fit value is f = 9.9 ± 2.8 × 10−16 m2/V2. To confirm this, dD/dt(D) was solved with this value of f for each of the dD/dt curves in Fig. 4 , showing excellent agreement [as also for additional dD/dt(D) profiles in ]. In particular, this is seen comparing the fits of Eq. to the data in Fig. 4 , with each pair taken at the same contact point (in independent experiments) but at different ψgold; crucially, the best fits to the experimental dD/dt(D) data are associated with the same f value for all four dD/dt(D) profiles, even though the uniform fields across the gap at D < ∼40 nm are significantly different. For comparison, the predicted curves according to Eq. but using f values corresponding to the highest and lowest literature estimates to date () are also shown in Fig. 4, showing clearly that they do not fit the data [similar misfits of these extremal f values to the data apply to the other dD/dt(D) profiles but are omitted for clarity]. Finally, in Fig. 4, we revisit the force profiles (F/R) versus D, Fig. 3, for which the discrepancy between theory and the forces evaluated from Eq. for D < ∼40 nm arose from ignoring the viscoelectric effect (i.e., setting f = 0 in Eq. : We see that once the directly determined value of f is inserted into Eq. , there is close agreement (within the scatter) between the evaluated forces and the theoretical prediction, further supporting our determination of the viscoelectric coefficient.

Conclusions

We have measured the viscoelectric effect in water by directly determining the changes in water viscosity under uniform electric fields through its modulation of the hydrodynamic damping between two approaching surfaces. Our result differs by up to one to two orders of magnitude from many of the earlier indirect estimates (7, 11, 12, 14–21), which were in the range (10−14–10−17) (m/V)2 () (though some of these indirect estimates, e.g., Refs. 7 and 8, were comparable to our measured one). Molecular dynamics simulation results suggest lower f values at fields much higher than in our study (4, 5, 40) and also that the relation of Eq. may not apply at such fields, while our results (Fig. 4) are consistent with Eq. . We do not know the reason for these discrepancies between experiment and MD simulations, though we should note that the fields used in the simulations could not be implemented over the length scales of water thickness in our experiments (10 to 30 nm), as electrolysis would occur (41); for this reason, comparison may not be appropriate. The main difference between our direct determination and previous estimates is that the latter were based on leveraging the electric field from the rapidly decaying potential gradient of double layers formed at a surface (or the field between two similar surfaces, which switches sign at the midplane due to symmetry). The analysis in those cases was based on an averaged electric field in the thin interfacial layer or in the intersurface gap (within which it varied very strongly), while in our study, the intersurface potential variation was very closely linear and its gradient—the electric field—therefore uniform across the water. We believe therefore that our measured value may be viewed as a direct measurement of the viscoelectric effect in water, conceptually similar to the direct measurements of this effect in polar organic liquids commenced over 80 y ago (1–3) but differing from all previous estimates of this effect in water, none of which involved measuring viscosity changes in uniform fields. It thus provides a benchmark of the viscoelectric effect, a basic phenomenon associated with coupling of the electric field to the dipole of water molecules, after decades of estimates covering a range of values differing by over a thousandfold. Improved characterization of the viscoelectric effect may shed light on many electrokinetic problems and help interpret boundary effects in aqueous environment, which has significant impact in real-life applications, as well as underpinning attempts at a better understanding of its molecular origins (4–6).

Materials and Methods

SFB.

As illustrated in Fig. 2, two plano-cylindrical glass lenses are arranged in a cross-cylinder configuration, with interaction geometry equivalent to a sphere approaching a flat surface. The upper lens is coated with a gold surface prepared through template stripping (30). The lower lens is covered by a single-crystal, back-silvered mica sheet. The upper lens is driven by the piezo crystal in a steady motion enabled by a function generator at a speed vapp < 10 nm/s toward the lower lens, mounted on a spring (constant k = 89 mN/mm), which bends in response to the total force between the surfaces. Normal force measurements were acquired from the bending of the spring k and by analyzing the hydrodynamics (Eq. . The three-electrode configuration (Fig. 2) enables positive potentials to be imposed onto the gold surface through a potentiostat when the platinum electrodes are positioned in the water immersing the surfaces, contained in a quartz bath to prevent short to ground. Further details of the technique are provided elsewhere (28, 32).

Preparation of Template-Stripped Gold Surface.

The gold surfaces were prepared by evaporating 99.999% pure gold pellet (Kurt J. Lesker and Materion) with Fabmated crucible onto atomistically smooth mica leaflets prepared by cleaving. The gold surface was glued to a cylindrical fused-silica lens (curvature radius R ∼1 cm) with epoxy resin (EPON Resin 1004F), and the mica leaflet was removed, leaving a gold surface of root mean-square roughness ∼0.2 nm (). Further details of the template-stripping method are given elsewhere (30).

Water Purification.

Water was purified with the Barnstead GenPure Pro Water Purification System, using high-performance ultraviolet assembly to reduce total organic content (TOC) to ultralow levels. The resistivity of the purified water is 18.2 MΩ ⋅ cm at 25 °C with TOC < 1 ppb.

Image Acquisition and Analysis.

A Hamamatsu Fusion C14440-20UP camera was set up to capture image sequence of FECO directly from the spectrometer without any lens in between. A calibration image was captured with a mercury (neon) calibration lamp, from which a linear relation between the pixel position and the wavelength was fitted. The image resolution is ∼0.4 Å in wavelength per pixel, and the best achieved resolution in surface-to-surface separation per pixel is 2.3 Å with mica leaflets thinner than 3 μm. During the jump-in events, images were acquired at 993 fps. The pixel position of the fringe tip, corresponding to the minimal distance between the lenses, was evaluated by fitting a double Gaussian function to the pixel intensity of the fringe doublet (Fig. 2). The separation D was evaluated using the multilayer matrix method, as described in previous works (30, 32, 42, 43).

Post–Data-Acquisition Processing.

To reduce the noise in the separation D(t) trajectory, multiple recordings were acquired consecutively at the same contact point for each applied gold surface potential. D(t) was smoothed with a moving average filter at 0.02 s intervals, chosen carefully to avoid artifacts from smoothing. dD(t)/dt was computed and averaged over repeated trajectories to reduce noise (). Note that, as demonstrated previously, charge inversion on the gold surface alone neutralizes the mica surface charge when in contact. Therefore, hysteresis observed in consecutive jump-in events of mica–mica systems due to unequilibrated charge release (44) is not observed in the gold–mica system (28).

Fitting Parameters in PB Equation.

The nonlinearized PB equation with constant potential on the gold surface and constant charge on the mica surface was solved numerically for more than 2,000 combinations of ψmica(D = ∞), ψgold(∀D), and bulk concentration cb, to sample ψmica(D = ∞) at 0.01 V intervals, ψgold(∀D) at 0.01 V intervals, and bulk concentration cb from 5 × 10−7 to 8 × 10−5 M at 1 × 10−6 M intervals. Each PB solution was augmented by van der Waals force FvdW(D) = AH/6D2 with Hamaker constant AH = 9 × 10−20 J (28, 32). The total force Ftot was compared to the normal force profile, i.e., the right-hand side of Eq. (η = η0), which was evaluated using the smoothed D(t) as described previously, averaged over repeated profiles, and smoothed with a 0.05 s interval moving average filter. Best fit was selected using the least mean-square deviation at large separations (D > 40 nm) where the viscoelectric effect is small, subject only to the constraint f ≥ 0. For each selected set of parameters, ψmica(D) and σgold(D) are evaluated for demonstrating charge inversion and for estimating the electric field E(D) at varying separations (Fig. 1 , and ). Uncertainties in these parameters manifest as uncertainties in the evaluated electric field and are indicated as error bars in the data of Fig. 4.

Numerical Solution to the Equation of Motion.

Eq. was made dimensionless by introducingwhich giveswhere the dimensionless number N0 indicates the relative scale of the surface forces compared to the viscous force. Eq. was solved numerically using the second-order Crank-Nicolson method (45), taking Ftot(D) and the corresponding E(D) as inputs. vapp was evaluated by linear fits to the experimentally measured D(t) at large separations D >200 nm, where Ftot and spring force are mostly absent. Parameters used were spring constant k = 89 N/m, lens radius R = 0.01 m, bulk water viscosity at room temperature η0 = 8.9 × 10−4 Pa ⋅ s, and initial separation D0 = 500 nm. To avoid spurious computational errors due to the exponentially growing |Ftot| as D decreases, the dimensionless time step Δtn* at each discrete step n was adjusted asand by updating the force scale F, the length scale L, and the velocity scale U such thatwhere the initial values are L0 = D0, U0 = vapp, and F0 = Ftot(D0).