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Hydrodynamic force between a sphere and a soft, elastic surface Farzaneh Kaveh, Javed Ally, Michael Kappl, and Hans-Jürgen Butt Langmuir, Just Accepted Manuscript • DOI: 10.1021/la502328u • Publication Date (Web): 08 Sep 2014 Downloaded from http://pubs.acs.org on September 13, 2014
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Langmuir
Hydrodynamic force between a sphere and a soft, elastic surface *
Farzaneh Kaveh, Javed Ally, Michael Kappl and Hans-Jürgen Butt Max Planck Institute for Polymer Research, Ackermannweg 10, 55128 Mainz, Germany
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Abstract
The hydrodynamic drainage force between a spherical silica particle and a soft, elastic polydimethylsiloxane surface was measured using the colloidal probe technique. The experimental force curves were compared to finite element simulations and an analytical model. The hydrodynamic repulsion decreased when the particle approached the soft surface as compared to a hard substrate. In contrast, when the particle was pulled away from the surface again, the attractive hydrodynamic force was increased. The hydrodynamic attraction increased because the effective area of the narrow gap between sphere and the plane on soft surfaces is larger than on rigid ones.
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Introduction
Wet bioadhesion allows animals such as tree frogs, limpets, and gastropods to stick to smooth surfaces even in wet or flooded conditions 1, 2, 3. For example, tree frogs adhere to surfaces by secreting a layer of mucus between their toe pads and the surface 4. Motivated by bioadhesion, we studied hydrodynamic forces between elastic, deformable surfaces. In biological systems, e.g. the bodies, foot pads, or feet of animals that use wet adhesion to stick to surfaces, the surfaces are soft and deform as they are pulled away from the substrate. Understanding wet bioadhesion would help to explain the biological adaptations that allow animals to stick to wet surfaces. In addition, it may help designing artificial biomimetic structures that replicate this ability. Such structures could be useful in a variety 1
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of applications, e.g. improving tire traction in wet conditions 5, or development of medical devices that can move or be fastened within the body. In wet adhesion, the space between the two adhering solid surfaces is filled with aqueous electrolyte. Wet adhesion is dominated by capillary and hydrodynamic forces; the latter is sometimes called Stefan adhesion 6. Van der Waals forces between solid surfaces are usually weak in liquids because the Hamaker constant is typically ten times lower than in air or vacuum
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. Capillary forces depend on the presence of a meniscus. Without a meniscus,
when the whole space is filled with liquid, they are absent. Stefan adhesion is the adhesive force that occurs due to viscous resistance as two parallel planar surfaces are pulled apart in a fluid. As two surfaces are pulled apart, the fluid has to fill the widening gap leading to a hydrodynamic viscous force. The smaller the gap and the higher the separation speed, the greater the hydrodynamic force required to separate the surfaces. Thus, Stefan adhesion is particularly important when pulling apart surfaces that are initially in mutual contact. In bioadhesion, this can help animals such as limpets
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to hold on to surfaces against violent
buffeting or shaking. Hydrodynamic forces have been measured and calculated for various geometries, such as two parallel circular plates 6, 10. The most important geometry is a sphere of radius R moving towards or away from a plane 11, 12, 13, 14. A sphere at a distance h from a plane moving at a velocity U in a Newtonian liquid of viscosity η experiences a distance dependent hydrodynamic force 7, 15. ܴ = ܨ−6ߨߟܴܷ ൬1 + ൰ ℎ
(1)
Equation (1) describes the hydrodynamic drainage force between sphere and plane and includes hydrodynamic Stokes friction of the sphere in the liquid. For distances smaller than the sphere radius, the drainage force is by far dominating. The negative sign indicates that the direction of the hydrodynamic force is opposite to the direction of the velocity. The first experiments on force between a sphere and a plane were carried out by observing a falling sphere
16, 17, 18
. In the last decade, hydrodynamic forces have been measured
extensively with the colloidal probe technique mainly to analyze slip 19, 20, 21, 22. Hydrodynamic forces have also been studied for film drainage between hard and liquid interfaces in the 2
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context of particle-bubble, particle-drop, or drop-drop interaction with the surface force apparatus (SFA)
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and the atomic force microscope (AFM)
24, 25
. Also hydrodynamic forces
between biomembranes have been analyzed 26. In contrast to experiments between rigid, undeformable surfaces, few experiments have been carried out between soft, elastic bodies27,
28
. Leroy and Charlaix studied elasto-
hydrodynamic interactions between oscillating spheres and elastic films, and developed an analytical model to determine the elastic modulus of the film29. This model was used to measure the elastic moduli of various substrates based on measurements with a SFA30 and to study drag force and slip length on microstructured surfaces31. The model of Leroy and Charlaix requires that the substrate deformation and amplitude of motion of the sphere are small relative to the distance between the sphere and substrate. Villey et al. demonstrated that even for seemingly rigid surface elastic deformation influences the hydrodynamic force once the separation reaches few nm
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. For studies relating to bioadhesion, we are,
however, interested in large amplitude motions that approach the point of contact between surfaces. In order to understand the effect of elastic deformability on hydrodynamic forces, we studied the interaction of a hard sphere and soft, deformable surface as they are brought into contact and pulled apart at various speeds using an AFM. We also developed a numerical finite element model of a sphere interacting with a soft substrate to study the effect of various parameters such as particle size and speed, liquid viscosity, and substrate elasticity. The experimental measurements were used to verify the accuracy of the finite element model. Using the verified numerical model, it was possible to study the effects of the parameters over larger ranges than feasible by experimental measurement alone. The numerical model provides additional information about the geometry of the deformed substrate and the liquid flow field around the particle, which cannot be determined from the experiments alone.
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Numerical model
Numerical simulation of the particle and soft substrate is complicated by the coupling between the liquid motion caused by the moving particle and the deformation of the 3
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substrate. When the particle is very close to the substrate, the deformation of the solid alters the boundary geometry of the liquid. We account for this coupling in our simulation. We consider the mechanics of the solid and liquid systems separately, and take advantage of the linearity of the equations governing their motion to couple the deformation of the substrate with the liquid flow.
Figure 1: The axisymmetric model geometry for a spherical solid particle approaching a solid substrate in a liquid environment. Position d and separation h are indicated. Considering a spherical particle moving normal to a surface allowed us to simplify our model to an axisymmetric geometry consisting of the section of the particle closest to the substrate, the fluid volume around this section of the particle, and the solid substrate (Figure 1). Since the particle is small we assume laminar flow. We do not consider the entire particle, as the distance-dependent viscous drag is dominated by the lower cap of the particle. Since the deformation of the substrate is small we assume a linear elastic response of the substrate over each time step of our simulation. As the particle is much more rigid than the substrate, deformation of the particle was ignored. We calculate the liquid flow and solid substrate deformation using finite element analysis (FEA) with the COMSOL Multiphysics package, version 4.2.a. We simulate the approach of the particle toward the substrate and its retraction. To model the particle motion, a no-slip condition33
was assumed at the liquid-particle and liquid-substrate interfaces. The
continuity equation was applied to simulate liquid flow, since the liquid was incompressible under our conditions.
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The base of the solid volume is rigidly constrained, and the stress and deformation at the edges of the solid volume are assumed to be zero since they are far from the applied stress due to the particle. This assumption and the volume size was checked and confirmed by convergence analysis by increasing the volume size until the difference in the total stress at the particle surface between successive runs was less than 1%. In the first step we simulated the liquid flow between the particle and substrate when the distance between them was larger than the particle diameter. The initial deformation of the substrate was negligible. The normal hydrodynamic pressure distribution in the liquid at the liquid-substrate interface was applied to the substrate. The radial shear stress is negligible, so it was not included in our model. We then calculated the substrate deformation under the applied hydrodynamic pressure. In the next step, the calculated deformation of the substrate was applied at the liquidsubstrate interface. To do this, we took the result of the initial step and, beginning from the center axis of the system where the deformation is the largest, found the point at which the deformation was less than 1%. The interval between the center axis and this point defines the deformed region of the liquid-substrate interface. The deformation was modeled with one hundred evenly spaced position data points in this region. The model geometry was updated by using a cubically interpolated polynomial curve defined by these hundred data points. The mesh was updated for the new geometry. After the initial step, the particlesubstrate separation, h, geometry and mesh were adjusted. The liquid flow was recalculated with the deformed geometry. The geometry was then updated for the next time step. This procedure was iterated until the particle was close to the substrate. The force on the particle at each step was calculated from the liquid flow in the simulation by integrating the stresses along the particle surface. The domain size in each simulation step was defined with respect to the separation distance between the probe and substrate.
The domain width was fifty times the separation
distance. The thickness of the solid and liquid domains were twenty times the separation distance. The mesh was defined such that there were at least 10 elements between the particle and substrate, with a maximum growth ratio of 1.1. In each time step, the geometry and mesh were updated and the Stokes equation was solved to determine the liquid flow. 5
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The resulting pressure distribution at the liquid-solid boundary was applied as a boundary condition for isotropic, linear elastic deformation of the solid domain. A consequence of this modeling approach is that the substrate deformation lags behind the velocity field; i.e. the substrate deformation in each time step corresponds to the liquid flow from the previous step. Thus, the time steps were chosen to be sufficiently short so that the difference can be neglected. In the experimental AFM measurement the motion of the particle is a combination of the constant velocity applied to the cantilever base and the cantilever deflection velocity due to the force on the particle. The additional shift in particle position due to cantilever deflection was taken into account in the model. The correction was determined in each time step by calculating the derivative of the force on the particle with time using a backward finite difference, and dividing this value by the spring constant of the cantilever used in the experiments. For the case of an AFM colloid probe approaching a non-deformable substrate, a direct numerical solution of equation (1) taking the slow-down of the particle close to the surface can be calculated without the need of a FEA simulation 20. We have carried out such direct calculations for comparison to validate our FEA simulations.
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Experimental Methods
As a model system, we measured the hydrodynamic interaction between borosilicate spheres (≈20 µm diameter) in ethanol and polydimethylsiloxane (PDMS) as a soft substrate. In addition, a silicon wafer was used as a hard substrate in order to have a reference for the PDMS deformation. The borosilicate particles were glued to the ends of AFM cantilevers (Mikromasch NSC-12 rectangular silicon cantilevers, length 130 μm, width 35 μm, 2 μm thick) with an epoxy glue (Epikote 1004) to make colloidal probes
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. Actual particle
diameters were measured using a scanning electron microscope (SEM) and ImageJ software version 1.46r, National Institutes of Health, USA. The colloidal probes and the silicon substrates were cleaned with ethanol and by plasma cleaning in argon for 10 min. The PDMS samples (Dow Sylgard 184) were prepared by mixing at a 10:1 ratio of elastomer-tocuring agent then degassing in a vacuum oven at room temperature. 3 mL of freshly prepared PDMS mixture were poured onto a clean silicon wafer and cured for one hour at 6
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90°C. Afterwards the PDMS was rinsed excessively with ethanol, dried and the force experiments were started without delay. The experiments were performed using an AFM equipped with a liquid cell (Multi Mode Picoforce, Veeco, USA) operated in closed-loop scanning mode. The liquid cell components (cantilever holder, O-rings etc.) were cleaned before each experiment in an ultrasonic bath with ethanol for two hours. For each measurement, a colloidal probe and sample were mounted in the AFM with the liquid cell, which was then filled with ethanol (anhydrous, Merck). We used ethanol instead of water to avoid hydrophobic interactions and reduce electrostatic double-layer repulsion. Force measurements were carried out after approaching the cantilever close to the substrate and allowing the system to equilibrate for 5 min. During the force measurements, the colloidal probe was moved into and out of contact with the substrate by the piezoelectric actuator of the AFM. The position of the piezo and the deflection of the cantilever were recorded continuously during the experiments using a data acquisition system (National Instruments PCI-6251) at a sampling frequency of 100 kHz. The velocities of the cantilever’s approach to and retraction from the substrate were equal, and ranged from 10 to 100 µm/s. The time between changing from approach to retraction was negligible. The recorded cantilever deflection signals were converted to force by multiplying with the system sensitivity and cantilever spring constant. The sensitivity was measured by slowly (2 µm/s) pressing the colloidal probe against a silicon wafer (Figure S1). Nominal spring constants of the cantilevers were 4.5 N/m; exact values were measured using the thermal noise method
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. The position of the colloidal probe was determined by subtracting the
cantilever deflection from the cantilever position. The resulting datasets consisted of the force on the particle as a function of its position d. Here we define zero position as the height, where the particle is contacting the substrate when the substrate is not deformed. When hydrodynamic forces deform the substrate, the position is zero where the undeformed substrate had been. The separation between substrate and particle is denoted by h (Figure 1).
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Results and Discussion
Hard substrate. As a first measurement, force curves were recorded with a low speed (2 µm/s). At such low speeds hydrodynamic forces were negligible. In these experiments a “contact part” was observed, characterized by a linear increase of the force with piezo position (Error! Reference source not found.1). Assuming that in this linear part real contact was established between the particles and the substrate we determined the sensitivity from the slope. In addition, zero position was defined at the intersection of this linear part with the zero force line. Zero positon was calculated for the approaching and retracting part separately. Typically both curvces were slightly offset most likely due to friction when the sphere on contact slides over the silicon wafer 36. On a a silicon wafer, the hydrodynamic force appears as a repulsion on approach (Figure 2a) and an attraction when retracting the sphere (Figure 2b). It decreased steeply with distance and increased with the velocity. Experiments were performed with several particles, cantilevers, and substrates. For better comparison, the experimental results shown in the following figures are taken from measurements with a single colloidal probe of 15.5 μm diameter on a cantilever with spring constant 1.87 N/m. The results obtained with other probes were consistent with those in Figure 2 and Figure 3. In all force curves the maximal force applied was constant (1 µN). Going to higher loads would have led to lower resolution due to the limited dynamic range. When an AFM cantilever moves through a fluid, whether or not a particle is attached, there is some drag force exerted on the cantilever itself. To account for hydrodynamic drag on the cantilever, the numerically simulated force curves were offset along the force axis. The required offset was determined from the measured force curves at each velocity when the particle was at maximum distance far away from the substrate for both approach and retraction. At this distance, the force is fully dominated by the hydrodynamic drag force on the cantilever. For comparing simulations and experiments, this cantilever drag force was added to the respective simulated force curves.
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Figure 2: Force curves for a 15.5 μm diameter silica particle (a) approaching and (b) retracting from a silicon surface. The solid lines are the experimental results; the dashed lines are the results of the numerical model, the dots correspond to the results calculated with Eq. (1). In all measurements on hard surfaces the maximum force was reached before the particle made actual contact with the surface. If the particle were in contact with the hard surface in any of the measurements, the slope of the force curve would be vertical; i.e. the force would increase as the particle is pushed onto the silica wafer without corresponding change in position. In all measurements the slope of the force curve was finite, indicating that the particle never actually made contact with the surface. In the absence of a vertical contact part in the force curves it was not possible to determine an absolute zero distance in the AFM measurements. Therefore, the offset in the position axis was determined using least-squares fit with the results of Eq. (1). With the abovementioned corrections, the numerically calculated force curves for a hard surface agreed with the measurements and forces calculated with Eq. (1). This agreement shows that the numerical model is valid for hard surfaces in the speed range explored.
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Force curves recorded at high speeds exhibit a lower attractive force upon retraction than expected by the model calculations. This can be explained by the fact that when approaching fast, the particle does not get as close to the substrate as it does when approaching slowly. As a result, the hydrodynamic drainage force is reduced because the starting separation (for retraction) is larger. Although experimental and theoretical force curves largely agreed, the experimental force curves were slightly steeper close to contact, whereas at large distances the experimental force was slightly lower than the calculated one. A possible explanation is the influence of the cantilever. We neglected any distance dependent contribution of the cantilever. There may be two contributions. First, the direct distance dependent hydrodynamic force on the cantilever. For spheres with R ≥5-7 µm this effect should be negligible because the cantilever is always at least 10-14 µm away from the substrate
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. Second, when the silica sphere
approaches the planar surface it is gradually slowed down. This slow-down not only reduces the hydrodynamic drag on the microsphere, but also on the cantilever
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. It effectively
reduced the hydrodynamic drag on the cantilever because the drag is proportional to the speed.
Soft substrates. For separations
