Reliable measurements of interfacial slip by colloid probe atomic force microscopy. I. Mathematical modeling

We developed a stable spread-sheet algorithm for the calculation of the hydrodynamic forces measured by colloid probe atomic force microscopy to be used in investigations of interfacial slip. The algorithm quantifies the effect on the slip hydrodynamic force for factors commonly encountered in experimental measurements such as nanoparticle contamination, nonconstant drag force due to cantilever bending that varies with different cantilevers, flattening of the microsphere, and calibration at large separations. We found that all of these experimental factors significantly affect the fitted slip length, approximately in the order listed. Our modeling is applied to fit new experimental data reproducibly. Using this new algorithm, it is shown that the fitting of hydrodynamic theories to experimental data is reliable and the fitted slip length is accurate. A "blind test" protocol was developed that produces a reliable estimate of the fitting error in the determination of both the slip length and spring constant. By this blind test, we estimate that our modeling determines the fitted slip length with an average systematic error of 2 nm and the fitted spring constant with a 3% error. Our exact calculation of the drag force may explain previous reports that the fitted slip length depends upon the shape and spring constant of the cantilever used to perform the measurements.     

Reliable measurements of interfacial slip by colloid probe atomic force microscopy. III. Shear-rate-dependent slip

We present experimental evidence and theoretical models that demonstrate that the slip length, which is the departure from the hydrodynamic no-slip boundary condition, cannot be constant as commonly assumed, but must decrease with increasing shear rate to avoid an unphysical divergence in the velocity of the fluid adjacent to the surface at small separations. The molecular origin of the shear rate dependence of the slip length is discussed. A new theoretical model for slip (the saturation model) is obtained, and it is shown to describe accurately colloid probe atomic force microscopy force measurements for all separations down to a few nanometers in two partially wetting situations (di-n-octyl phthalate on silanized silicon and bare silicon). Previous observations of slip length increasing with shear rate are explained as due to an imprecise calculation of the drag force on the cantilever. A new way of plotting experimental data is also presented, which provides a useful way to illustrate the slip length dependence on the shear rate.     

Reconciling slip measurements in symmetric and asymmetric systems

In the past decade, the slip of simple liquids on solid surfaces has been demonstrated by many groups. However, the slip of liquids on wettable surfaces is heavily debated. Using colloid probe atomic force microscopy (AFM), we found the slip length of di-n-octylphthalate in a symmetric wettable system (silica) to be around 11 nm, which raises the question of what the measured slip length in an asymmetric hydrophilic-hydrophobic system would be. To answer this question, we investigated liquid slip in one symmetric nonwettable system (hydrophobic DCDMS or OTS) and in one asymmetric hydrophilic (silica)-hydrophobic (DCDMS) system by the same method at driving velocities of between 10 and 80 μm/s. The slip results obtained from the three systems are in agreement with each other, and this comparison provides a means to self-assess the accuracy and reproducibility of the measured force curves and the fitted slip length in our systems. Furthermore, this method provides access to reliable values of the actual slip length on any investigated flat surface in an asymmetric system, avoiding the difficulty of preparing a symmetric probe/flat surface system in a colloid probe AFM force measurement.     

Measurement of dynamical forces between deformable drops using the atomic force microscope. I. Theory

Recent experimental developments have enabled the measurement of dynamical forces between two moving liquid drops in solution using an atomic force microscope (AFM). The drop sizes, interfacial tension, and approach velocities used in the experiments are in a regime where surface forces, hydrodynamics, and drop deformation are all significant. A detailed theoretical model of the experimental setup which accounts for surface forces, hydrodynamic interactions, droplet deformation, and AFM cantilever deflection has been developed. In agreement with experimental observations, the calculated force curves show pseudo-constant compliance regions due to drop flattening, as well as attractive pull-off forces due mainly to hydrodynamic lubrication forces.     

Atomic force microscopy colloid-probe measurements with explicit measurement of particle-solid separation

We describe the use of evanescent wave scattering to measure the separation between the surface of a solid and a particle that is attached to an atomic force microscope (AFM) cantilever. Termed evanescent wave atomic force microscopy, our approach involves measuring the intensity of the light scattered from an evanescent field formed by the total internal reflection of a laser beam at a solid/fluid interface. In a conventional AFM "colloid probe" measurement, this separation must be inferred from an examination of the surface forces. Direct measurement of this separation with an evanescent wave atomic force microscope (EW-AFM) removes some ambiguity in the surface force measurement and, in addition, allows new types of measurements. For example, the force can be monitored at a constant separation. Our evanescent scattering apparatus is essentially identical to that used in total internal reflection microscopy (TIRM), except that we collect the light that scatters back into the incident medium, because the AFM partly obscures the forward scattered light (i.e., light scattered into the transmitted region). Compared to a conventional TIRM measurement, where the particle moves freely, attaching the particle to the cantilever in an EW-AFM gives much greater control of the particle position.     

Elastic deformation during dynamic force measurements in viscous fluids

Understanding and harnessing the coupling between lubrication pressure and elasticity provides materials design strategies for applications such as adhesives, coatings, microsensors, and biomaterials. Elastic deformation of compliant solids caused by viscous forces can also occur during dynamic force measurements in instruments such as the surface forces apparatus (SFA) or the atomic force microscope (AFM). We briefly review hydrodynamic interactions in the presence of soft, deformable interfaces in the lubrication limit. More specifically, we consider the scenario of two surfaces approaching each other in a viscous fluid where one or both surfaces is deformable, which is also relevant to many force measurement systems. In this article the basic theoretical background of the elastohydrodynamic problem is detailed, followed by a discussion of experimental validation and considerations, especially for the role of elastic deformation on surface forces measurements. Finally, current challenges to our understanding of soft hydrodynamic interactions, such as the consideration of substrate layering, poroelasticity, viscoelasticity, surface heterogeneity, as well as their implications are discussed.     

Settling of a charged hydrophobic rigid colloid in aqueous media under generalized gravitational field

The hindrance created by the induced electric filed on the sedimentation of a charged colloid in an aqueous media is studied through numerical modeling. The colloid is considered to be hydrophobic, sedimenting under gravity or a centrifugal force (generalized gravity). The deformation of the charge cloud around the colloid induces an electric field, which generates electrical dipole force on the colloid. The sedimentation velocity is governed by the balance of an electric force, hydrodynamic drag, and gravitational force. Governing equations based on the first principle of electrokinetics is solved numerically through a control volume approach. The dependence of the sedimentation velocity on the electrical properties and slip length of the colloid is investigated. The sedimentation velocity of the charged colloid is slower than the corresponding uncharged particle and this deviation magnifies as the charge density as well as particle slip length is increased. An enhanced g-factor creates a size dependency of the charged colloids. The induced sedimentation field is obtained to analyze the electrokinetics. Surface hydrophobicity enhances the sedimentation velocity, which in turn manifests the induced sedimentation field. However, the sedimentation velocity of a charged hydrophobic colloid is lower than the corresponding uncharged hydrophobic particle and this deviation manifests as slip length is increased.     

Direct comparison of atomic force microscopic and total internal reflection microscopic measurements in the presence of nonadsorbing polyelectrolytes

We have investigated the structural and depletion forces between silica glass surfaces in aqueous, salt-free solutions of sodium poly(styrene sulfonate). The interaction forces were investigated by two techniques: total internal reflectance microscopy (TIRM) and colloid probe atomic force microscopy (AFM). The TIRM technique measures the potential energy of interaction directly, while the AFM is a force balance. Comparison between the data sets was used to independently calibrate the AFM data since the separation distances cannot be unequivocally determined by this technique. Oscillatory structural forces are excellent for this work since they give multiple reference points against which to analyze. Comparison of the data from the two techniques highlighted significant uncertainties in the AFM data. At low polymer concentrations, a significant uncertainty in the apparent zero separation distance was seen as a result of the AFM cantilever reaching an apparent constant compliance region prior to any real contact between the surfaces. Further complications arising from the number and position of the measured minima were also seen in the dilute polymer concentration regime as a result of hydrodynamic drainage between the approaching surfaces in the AFM perturbing the delicate structural components in the fluid.     

Analysis of bacterial adhesion using a gradient force analysis method and colloid probe atomic force microscopy

The atomic force microscope (AFM) has been used to examine the stickiness of bacteria on the basis of the analysis of approach and retraction force curves between the AFM tip and the bacterial surface. One difficulty in analyzing approach curve data is that the distance between the AFM tip and the surface of the bacterium is difficult to define. The exact distances are difficult to determine because the surface of the bacterium deforms during force imaging, producing a highly nonlinear region in the approach curve. In this study, AFM approach and retraction curves were obtained using a colloid probe AFM for three strains of Escherichia coli (D21, D21f2, and JM109). These strains differed in their relative adhesion to glass surfaces, on the basis of measurements of sticking coefficients in packed bed flow through column tests. A gradient force curve analysis method was developed to model the interactions between the colloid probe and a surface. Gradient analysis of the approach curve revealed four different regions of colloid-surface interactions during the approach and contact of the probe with the bacterial surface: a noninteraction region, a noncontact phase, a contact phase, and a constant compliance region. The noncontact phase, which ranged from 28 to 59 nm for the three bacterial strains, was hypothesized to arise primarily from steric repulsion of the colloid by extracellular polymers on the bacterial surface. The contact phase, spanning 59-113 nm, was believed to arise from the initial pressure of the colloid on the outer membrane of the cell. The constant compliance region likely reflected the response of the colloid probe to the stiff peptidoglycan layer that confers strength and rigidity to gram negative bacteria. It was shown that the sticking coefficients reported for the three E. coli strains were correlated with the length of the noncontact phase but not the properties of the other phases. Sticking coefficients were also not correlated with any parameters determined from retraction force curves such as pull-off distances or separation energies. These results show that gradient analysis is useful for studying the contribution of the length of the exopolymers on the cell surface to bacterial adhesion to glass surfaces.     

Novel and global approach of the complex and interconnected phenomena related to the contact line movement past a solid surface from hydrophobized silica gel

In this work a generalized hydrodynamic theory for water flow into a mesoporous matrix from hydrophobized silica gel is suggested. Although we examine a fluid dynamics problem, the motion of the water-gas-solid contact line past a hydrophobized silica gel surface, motivation for such research derives from the investigation of a novel principle of mechanical energy dissipation, called surface dissipation, and its attached machine element, named a colloidal damper (CD). Similar to a hydraulic damper, this absorber has a cylinder-piston structure, but oil is replaced by a colloid consisting of a mesoporous matrix and a lyophobic liquid. Here, the mesoporous matrix is from silica gel modified by linear chains of alkyldimethylchlorosilanes and water is the associated lyophobic liquid. Mainly, the colloidal damper energy loss can be explained by the dynamic contact angle hysteresis in advancing (liquid displaces gas) and receding motion (gas displaces liquid); such hysteresis occurs due to the geometrical and chemical heterogeneities of the solid surface. Although this new kind of dissipation could be attractive for many applications, the subject remains almost unexplored in the scientific literature. Many different, complex, and interconnected aspects are related to this subject: capillary hydrodynamics, slippage effect, contact angle hysteresis, estimation of dissipated energy, thickness optimization of the grafted layer on the surface of the mesoporous matrix, etc. For this reason, a novel and global approach to all the complex and interconnected phenomena related to the contact line movement past a solid surface from hydrophobized silica gel is proposed. Our approach has a modest experimental basis but this is compensated for with rich references to other experimental and theoretical work oriented to the study of surface phenomena in such systems. We tried to sort the existing results and to find the right place for each in building our global view of the problem. This work is structured as follows. The measurement technique of the hysteresis loop is described. From experimental data one calculates the dissipated energy versus length of the grafted molecule on the silica gel surface. These results are justified by flow analysis. Generalized hydrodynamic theory means here that the basic structure of the Navier-Stokes equations is kept, but in order to include the relation between macroscopic flow and molecular interactions, slip is allowed on the solid wall. The nanopillar architecture of the silica gel hydrophobic coating is described. Concepts of slip and contact angle hysteresis are detailed and their connection is revealed. During adsorption, water penetrates the pore space by maintaining contact with the top of the coating molecules (region of -CH(3) groups); after that, water is forced into and partially or totally fills the space between molecules (region of -CH(2) groups). In such circumstances, at the release of the external pressure, desorption occurs. An original energetic-barriers approach is proposed to understand the filling of the nanosize canals which occur in the hydrophobic grafted layer. Employing this energetic-barriers approach, one finds the optimum length of the grafted molecule which maximizes the dissipated energy of the CD reversible cycle. Such results are useful for the appropriate design of ultrahydrophobic surfaces in general, and for the optimal design of a hydrophobic coating of a mesoporous matrix destined for CD use.     

Imaging of a soft, weakly adsorbing, living cell with a colloid probe tapping atomic force microscope technique

Here, we propose a new method to improve the atomic force microscopy (AFM) image resolution of soft samples, such as cells, in liquid. Attaching a colloid probe to a cantilever was seen improve the image resolution of a living cell in a physiological buffer solution, obtained by the normal tapping mode, when compared to an image obtained using a regular cantilever tip. This may be due to the averaging out of the cantilever tip swinging caused by the visco-elasticity of the cell. The resolution was best, when silica spheres with a 3.3 microm diameter were attached. Although larger spheres gave a resolution better than a bare cantilever tip, their resolution was less than that obtained for the 3.3 microm diameter silica colloid. This dependency of the image resolution on the colloid probe size may be a result of the increased macroscopic van der Waals attraction between the cell and probe, the decreased repulsive force dependence on the cantilever probe radius, and the decrease in resolution due to the increased probe size. The size of the colloid probe, which should be attached to the cantilever to give the best image resolution, would be the one that optimises the combined result of these facts.     

Theory of non-equilibrium force measurements involving deformable drops and bubbles

Over the past decade, direct force measurements using the Atomic Force Microscope (AFM) have been extended to study non-equilibrium interactions. Perhaps the more scientifically interesting and technically challenging of such studies involved deformable drops and bubbles in relative motion. The scientific interest stems from the rich complexity that arises from the combination of separation dependent surface forces such as Van der Waals, electrical double layer and steric interactions with velocity dependent forces from hydrodynamic interactions. Moreover the effects of these forces also depend on the deformations of the surfaces of the drops and bubbles that alter local conditions on the nanometer scale, with deformations that can extend over micrometers. Because of incompressibility, effects of such deformations are strongly influenced by small changes of the sizes of the drops and bubbles that may be in the millimeter range. Our focus is on interactions between emulsion drops and bubbles at around 100 μm size range. At the typical velocities in dynamic force measurements with the AFM which span the range of Brownian velocities of such emulsions, the ratio of hydrodynamic force to surface tension force, as characterized by the capillary number, is ~ 10^{− 6} or smaller, which poses challenges to modeling using direct numerical simulations. However, the qualitative and quantitative features of the dynamic forces between interacting drops and bubbles are sensitive to the detailed space and time-dependent deformations. It is this dynamic coupling between forces and deformations that requires a detailed quantitative theoretical framework to help interpret experimental measurements. Theories that do not treat forces and deformations in a consistent way simply will not have much predictive power. The technical challenges of undertaking force measurements are substantial. These range from generating drop and bubble of the appropriate size range to controlling the physicochemical environment to finding the optimal and quantifiable way to place and secure the drops and bubbles in the AFM to make reproducible measurements. It is perhaps no surprise that it is only recently that direct measurements of non-equilibrium forces between two drops or two bubbles colliding in a controlled manner have been possible. This review covers the development of a consistent theory to describe non-equilibrium force measurements involving deformable drops and bubbles. Predictions of this model are also tested on dynamic film drainage experiments involving deformable drops and bubbles that use very different techniques to the AFM to demonstrate that it is capable of providing accurate quantitative predictions of both dynamic forces and dynamic deformations. In the low capillary number regime of interest, we observe that the dynamic behavior of all experimental results reviewed here are consistent with the tangentially immobile hydrodynamic boundary condition at liquid–liquid or liquid–gas interfaces. The most likely explanation for this observation is the presence of trace amounts of surface-active species that are responsible for arresting interfacial flow.     

Adhesion characteristics of two Burkholderia cepacia strains examined using colloid probe microscopy and gradient force analysis

Colloid probe atomic force microscopy (CP-AFM) was used to investigate two strains of Burkholderia cepacia in order to determine what molecular scale characteristics of strain Env435 make it less adhesive to surfaces than the parent strain, G4. CP-AFM approach curves analyzed using a gradient force method showed that in a high ionic strength solution (IS=100 mM, Debye length=1 nm), the colloid probe was attracted to the surface of strain G4 at a distance of approximately 30 nm, but it was repelled over a distance of 25 nm when approaching strain Env435. Adhesion forces measured under the same solution conditions during colloid retraction showed that 1.38 nN of force was required to remove the colloid placed in contact with the surface of strain G4, whereas only 0.58 nN was required using strain Env435. At IS=1mM (Debye length=10nm), the attractive force observed with G4 was no longer present, and the repulsive force seen with Env435 was extended to approximately 250 nm. The adhesion of the bacteria to the probe was much less at low IS solution (1 mM) than at high IS (100 mM). The greater adhesion characteristics of strain G4 compared to Env435 were confirmed in column tests. Strain G4 had a collision efficiency of alpha=0.68, while strain Env435 had a much lower collision efficiency of alpha=0.01 (IS=100 mM). These results suggest that the reduced adhesion of strain Env435 measured in column tests is due to the presence of high molecular weight extracellular polymeric substances that extend out from the cell surface, creating long-range steric repulsion between the cell and a surface. Adhesion is reduced as these polymers do not appear to be "sticky" when placed in contact with a surface in AFM tests.     

In situ hydrodynamic lateral force calibration of AFM colloidal probes

Lateral force microscopy (LFM) is an application of atomic force microscopy (AFM) to sense lateral forces applied to the AFM probe tip. Recent advances in tissue engineering and functional biomaterials have shown a need for the surface characterization of their material and biochemical properties under the application of lateral forces. LFM equipped with colloidal probes of well-defined tip geometries has been a natural fit to address these needs but has remained limited to provide primarily qualitative results. For quantitative measurements, LFM requires the successful determination of the lateral force or torque conversion factor of the probe. Usually, force calibration results obtained in air are used for force measurements in liquids, but refractive index differences between air and liquids induce changes in the conversion factor. Furthermore, in the case of biochemically functionalized tips, damage can occur during calibration because tip-surface contact is inevitable in most calibration methods. Therefore, a nondestructive in situ lateral force calibration is desirable for LFM applications in liquids. Here we present an in situ hydrodynamic lateral force calibration method for AFM colloidal probes. In this method, the laterally scanned substrate surface generated a creeping Couette flow, which deformed the probe under torsion. The spherical geometry of the tip enabled the calculation of tip drag forces, and the lateral torque conversion factor was calibrated from the lateral voltage change and estimated torque. Comparisons with lateral force calibrations performed in air show that the hydrodynamic lateral force calibration method enables quantitative lateral force measurements in liquid using colloidal probes.     

Isoelectric point of fluorite by direct force measurements using atomic force microscopy

Interaction forces between a fluorite (CaF2) surface and colloidal silica were measured by atomic force microscopy (AFM) in 1 x 10(-3) M NaNO3 at different pH values. Forces between the silica colloid and fluorite flat were measured at a range of pH values above the isoelectric point (IEP) of silica so that the forces were mainly controlled by the fluorite surface charge. In this way, the IEP of the fluorite surface was deduced from AFM force curves at pH approximately 9.2. Experimental force versus separation distance curves were in good agreement with theoretical predictions based on long-range electrostatic interactions, allowing the potential of the fluorite surface to be estimated from the experimental force curves. AFM-deduced surface potentials were generally lower than the published zeta potentials obtained from electrokinetic methods for powdered samples. Differences in methodology, orientation of the fluorite, surface carbonation, and equilibration time all could have contributed to this difference.     

Sensitive quantitative nucleic acid detection using oligonucleotide microarrays

We report a new theoretical approach to optimize the performance and quantify the results of gene expression oligonucleotide microarrays which are widely used in biomedical research. An on-array hybridization isotherm that takes into account the screened Coulomb repulsion between the assayed nucleic acid target and the layer of surface tethered oligonucleotide probes is presented. The hybridization efficiency is found as a function of the genomic target (sequence, length, and concentration), array parameters (probe sequence and length, surface probe density), and hybridization conditions (temperature and buffer ionic strength). We present simple relations for the hybridization signal maximum and the linear dynamic detection range and show explicit criteria for optimization. The approach is based on an extension of our recently published theory (Vainrub, A.; Pettitt, B. M. Phys. Rev. E 2002, 66, art. no.-041905) which we generalize here for the cases of target depletion effects and arbitrary target length.     

Effects of flow on measurements of interactions in colloidal suspensions

A hydrodynamic mechanism of interactions of colloidal particles is considered. The mechanism is based on the assumption of tiny background flows in the experimental cells during measurements by Grier et al. Both trivial (shear flow) and nontrivial (force propagation through viscous fluid) effects are taken into account for two colloidal particles near a wall bounding the solvent. Expressions for the radial (attractive or repulsive) forces and the polar torques are obtained. Quantitative estimates of the flow needed to produce the observed strength of attractive force are given; other necessary conditions are also considered. The following conclusion is made: the mechanism suggested most likely is not responsible for the attractive interactions observed in the experiments of Grier et al.; however, it may be applicable in other experimental realizations and should be kept in mind while conducting colloidal measurements of high sensitivity. Several distinctive features of the interactions due to this mechanism are identified.     

Direct measurement of depletion and hydrodynamic forces in solutions of a reversible supramolecular polymer

In this paper, the investigation of surface forces in semidilute solutions of a nonadsorbing hydrogen-bonded reversible supramolecular polymer is described. Colloidal probe atomic force microscopy was used for direct measurement of depletion forces. Hydrodynamic drag on the AFM cantilever with the colloidal probe was measured both far away from and close to the planar substrate surface. The results indicate that the presence of the depletion layer causes slip at the surfaces with a large apparent slip length. Our analysis explains how the presence of slip enables the measurement of (relatively weak) depletion forces in solutions with a high viscosity by significantly reducing the hydrodynamic forces. The range and magnitude of the measured depletion forces are qualitatively in agreement with previous experiments and theoretical predictions. Due to the relatively large experimental error, no quantitative conclusions can be drawn. Depletion-induced phase separation of suspensions of stearylated silica particles was also observed. Phase separation becomes more pronounced with increasing polymer concentration.     

Parameters affecting the adhesion strength between a living cell and a colloid probe when measured by the atomic force microscope

In this study, we used the colloid probe atomic force microscopy (AFM) technique to investigate the adhesion force between a living cell and a silica colloid particle in a Leibovitz's L-15 medium (L-15). The L-15 liquid maintained the pharmaceutical conditions necessary to keep the cells alive in the outside environment during the AFM experiment. The force curves in such a system showed a steric repulsion in the compression force curve, due to the compression of the cells by the colloid probe, and an adhesion force in the decompression force curve, due to binding events between the cell and the probe. We also investigated for the first time how the position on the cell surface, the strength of the pushing force, and the residence time of the probe at the cell surface individually affected the adhesion force between a living cell and a 6.84 μm diameter silica colloid particle in L-15. The position of measuring the force on the cell surface was seen not to affect the value of the maximum adhesion force. The loading force was also seen not to notably affect the value of the maximum adhesion force, if it was small enough not to pierce and damage the cell. The residence time of the probe at the cell surface, however, clearly affected the adhesion force, where a longer residence time gave a larger maximum force. From these results, we could conclude that the AFM force measurements should be made using a loading force small enough not to damage the cell and a fixed residence time, when comparing results of different systems.