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.     

Hydrodynamic interaction between spheres coated with deformable thin liquid films

In this article, we considered the hydrodynamic interaction between two unequal spheres coated with thin deformable liquids in the asymptotic lubrication regime. This problem is a prototype model for drop coalescence through the so-called "film drainage" mechanism, in which the hydrodynamic contribution comes dominantly from the lubrication region apart from the van der Waals interaction force. First, a general formulation was derived for two unequal coated spheres that experienced a head-to-head collision at a very close proximity. The resulting set of the evolution equations for the deforming film shapes and stress distributions was solved numerically. The film shapes and hydrodynamic interaction forces were determined as functions of the separation distance, film thickness, viscosity ratios, and capillary numbers. The results show that as the two spheres approach each other, the films begin to flatten and eventually to form negative curvature (or a broad dimple) at their forehead areas in which high lubrication pressure is formed. The dimple formation occurs earlier as the capillary number increases. For large capillary numbers, the film liquids are drained out from their forehead areas and the coated liquid films rupture before the two films "touch" each other. Meanwhile, for small capillary numbers, the gap liquid is drained out first and the two liquid films eventually coalesce.     

Interactions between two bubbles on a hot or cold wall

A temperature gradient normal to a planar wall produces two-dimensional motion and aggregation or separation of bubbles on the hot or cold wall, respectively. The origin of the motion is fluid convection driven by the thermal Marangoni stress on the surface of the bubbles. Previous theories for the dynamics of two or more bubbles have been based on an analysis of flow about a single bubble and the resulting convection that entrains its neighbors. Here we extend the theory by solving the quasi-steady equations for the temperature and velocity fields for two bubbles. The result is a quantitative model for the relative velocity between two bubbles as a function of both the distance between them and the gap between each bubble and the surface. Interactions between the bubbles strongly increase the approach velocity, which is counter-intuitive because the hydrodynamic resistance increases as the bubbles approach each other. An asymptotic analysis indicates the thermocapillary force bringing them together or pushing them apart is singular in the separation when the bubbles are close to each other. The two-bubble theory agrees reasonably well with the experimentally measured velocities of pairs of bubbles on hot or cold surfaces, though it slightly overestimates the velocities.     

Experiments on the motion of drops on a horizontal solid surface due to a wettability gradient

Results from experiments performed on the motion of drops of tetraethylene glycol in a wettability gradient present on a silicon surface are reported and compared with predictions from a recently developed theoretical model. The gradient in wettability was formed by exposing strips cut from a silicon wafer to dodecyltrichlorosilane vapors. Video images of the drops captured during the experiments were subsequently analyzed for drop size and velocity as functions of position along the gradient. In separate experiments on the same strips, the static contact angle formed by small drops was measured and used to obtain the local wettability gradient to which a drop is subjected. The velocity of the drops was found to be a strong function of position along the gradient. A quasi-steady theoretical model that balances the local hydrodynamic resistance with the local driving force generally describes the observations; possible reasons for the remaining discrepancies are discussed. It is shown that a model in which the driving force is reduced to accommodate the hysteresis effect inferred from the data is able to remove most of the discrepancy between the observed and predicted velocities.     

Electroviscous Forces on a Charged Cylinder Moving Near a Charged Wall

The refined theory of the electroviscous lift forces is presented for the case when the separation distance between the particle and the wall is larger than the double-layer thickness. The theory is based on the lubrication approximation for motion of a long cylinder near a solid wall in creeping flow. The approximate analytical formula for the lift force valid for Pe相似文献   

Electroosmotic flow in a rectangular channel with variable wall zeta-potential: comparison of numerical simulation with asymptotic theory

Electroosmotic flow in a straight micro-channel of rectangular cross-section is computed numerically for several situations where the wall zeta-potential is not constant but has a specified spatial variation. The results of the computation are compared with an earlier published asymptotic theory based on the lubrication approximation: the assumption that any axial variations take place on a long length scale compared to a characteristic channel width. The computational results are found to be in excellent agreement with the theory even when the scale of axial variations is comparable to the channel width. In the opposite limit when the wavelength of fluctuations is much shorter than the channel width, the lubrication theory fails to describe the solution either qualitatively or quantitatively. In this short wave limit the solution is well described by Ajdari's theory for electroosmotic flow between infinite parallel plates (Ajdari, A., Phys. Rev. E 1996, 53, 4996-5005.) The infinitely thin electric double layer limit is assumed in the theory as well as in the simulation.     

Hydrodynamic forces acting on a microscopic emulsion drop growing at a capillary tip in relation to the process of membrane emulsification

Here, we calculate the hydrodynamic ejection force acting on a microscopic emulsion drop, which is continuously growing at a capillary tip. This force could cause drop detachment in the processes of membrane and microchannel emulsification, and affect the size of the released drops. The micrometer-sized drops are not deformed by gravity and their formation happens at small Reynolds numbers despite the fact that the typical period of drop generation is of the order of 0.1 s. Under such conditions, the flow of the disperse phase through the capillary, as it inflates the droplet, engenders a hydrodynamic force, which has a predominantly viscous (rather than inertial) origin. The hydrodynamic boundary problem is solved numerically, by using appropriate curvilinear coordinates. The spatial distributions of the stream function and the velocity components are computed. The hydrodynamic force acting on the drop is expressed in terms of three universal functions of the ratio of the pore and drop radii. These functions are computed numerically. Interpolation formulas are obtained for their easier calculation. It turns out that the increase in the viscosity of each of the two liquid phases increases the total ejection force. The results could find applications for the interpretation and prediction of the effect of hydrodynamic factors on the drop size in membrane emulsification.     

Simultaneous measurement of structural and hydrodynamic forces between colloidal surfaces in complex fluids

An AFM study was performed to measure the effect of approach/retraction speed on the interaction force between a colloidal particle and a flat substrate in aqueous solutions containing silica nanospheres at concentrations of 4.5 and 6.5 vol.%. The total force consisted of contributions of electrostatic, depletion, structural and hydrodynamic forces. The hydrodynamic component of the force could be isolated by comparing the force profiles measured upon approach and retraction. It was found that when the hydrodynamic component was subtracted from the total force, the resulting force profiles measured at scan speeds of 80, 800, 2400, 4800 and 11,200 nm/s all overlaid, indicating that the surface forces (electrostatic, depletion and structural) were not affected by the scan speed. This result was further supported by an approximation of the rates of viscous and diffusive motion in the gap region. In addition, the variation of the hydrodynamic force with particle/plate separation distance agreed relatively well with a prediction made using the mobility correction factor developed for simple fluids, suggesting that the nanoparaticles do not alter the flow in the lubrication layer at these concentrations.     

Deformation and motion of a charged conducting drop in a dielectric liquid under a nonuniform electric field

As a tool for transporting a drop inside another fluid, a charged conducting drop driven by Coulombic force is considered. Specifically, deformation and motion of a charged conducting drop under nonuniform electric fields are studied using the perturbation method. For simplicity in analysis, the applied electric field is assumed to be expressed as the sum of a uniform field and a linear field and the flow is assumed to be in the Stokes flow range. The deformed drop shape due to electrical stress is computed to the first order of the electrical Weber number (W). Then the electric force and the hydrodynamic drag are computed to derive the formula of the translation velocity, which is valid up to O(W). Several important results have also been obtained for the effect of drop deformation on the electric and hydrodynamic forces exerted on the drop.     

The drop size in membrane emulsification determined from the balance of capillary and hydrodynamic forces

Here, we investigate experimentally and theoretically the factors that determine the size of the emulsion droplets produced by membrane emulsification in "batch regime" (without applied crossflow). Hydrophilic glass membranes of pore diameters between 1 and 10 mum have been used to obtain oil-in-water emulsions. The working surfactant concentrations are high enough to prevent drop coalescence. Under such conditions, the size of the formed drops does not depend on the surfactant type and concentration, on the interfacial tension, or on the increase of viscosity of the inner (oil) phase. The drops are monodisperse when the working transmembrane pressure is slightly above the critical pressure for drop breakup. At higher pressures, the size distribution becomes bimodal: a superposition of a "normal" peak of monodisperse drops and an "anomalous" peak of polydisperse drops is observed. The theoretical model assumes that, at the moment of breakup, the hydrodynamic ejection force acting on the drop is equal to the critical capillary force that corresponds to the stability-instability transition in the drop shape. The derived equations are applied to predict the mean size of the obtained drops in regimes of constant flow rate and constant transmembrane pressure. Agreement between theory and experiment is established for the latter regime, which corresponds to our experimental conditions. The transition from unimodal to bimodal drop size distribution upon increase of the transmembrane pressure can be interpreted in terms of the transition from "dripping" to "jetting" mechanisms of drop detachment.     

Electrokinetic lift force for a charged particle moving near a charged wall — a modified theory and experiment

We present the refined theory of the electrokinetic lift force for a charged particle moving at a charged wall at the distance much larger than the double layer thickness. The theory is based on the lubrication approximation for the solution of the Stokes equation for the flow around a long cylinder moving near a solid wall. The “thin double layer” approximation is used to solve the ionic balance and electro-osmotic flow equations. The electrokinetic lift force is then obtained by integration of the viscous stress tensor as well as the Maxwell stress tensor over the particle surface. The resulting lift force for the cylinder translating, rotating at the wall as well as for the stationary cylinder in the wall shear flow, is considered. Following this, we apply the Derjaguin approximation to transform the obtained results to the sphere–wall geometry and we compare our theoretical predictions with the measurements of the electrokinetic lift force performed in the “colloidal particle collider” apparatus for the latex particles suspended in the glycerol–water solutions. Our theoretical results for the electrokinetic lift force exceeds by several orders of magnitude one obtained from the previously developed theory and are in a good agreement with experimental findings.     

Electroviscous cylinder-wall interactions

A theoretical analysis is presented to determine the forces of interaction between an electrically charged cylindrical particle and a charged plane boundary wall when the particle translates parallel to the wall and rotates around its axis in a symmetric electrolyte solution at rest. The electroviscous effects, arising from the coupling between the electrical and hydrodynamic equations, are determined as a solution of three partial differential equations, derived from R.G. Cox's general theory [J. Fluid Mech. 338 (1997) 1], for electroviscous ion concentration, electroviscous potential, and electroviscous flow field. It is assumed a priori that the double layer thickness surrounding each charged surface is much smaller than the length scale of the problem. Using the matched asymptotic expansion technique, the electroviscous forces experienced by the cylinder are explicitly determined analytically for small particle-wall distances for low and intermediate Peclet numbers. It is found that the tangential force usually increases the drag above the purely hydrodynamic drag, although for certain conditions the drag can be reduced. Similarly the normal force is usually repulsive, i.e., it is an electrokinetic lift force, but under certain conditions the normal force can be attractive.     

Effect of small particles on the near-wall dynamics of a large particle in a highly bidisperse colloidal solution

We consider the hydrodynamic effect of small particles on the dynamics of a much larger particle moving normal to a planar wall in a highly bidisperse dilute colloidal suspension of spheres. The gap h(0) between the large particle and the wall is assumed to be comparable to the diameter 2a of the smaller particles so there is a length-scale separation between the gap width h(0) and the radius of the large particle b>h(0). We use this length-scale separation to develop a new lubrication theory which takes into account the presence of the smaller particles in the space between the larger particle and the wall. The hydrodynamic effect of the small particles on the motion of the large particle is characterized by the short time (or high frequency) resistance coefficient. We find that for small particle-wall separations h(0), the resistance coefficient tends to the asymptotic value corresponding to the large particle moving in a clear suspending fluid. For h(0)>a, the resistance coefficient approaches the lubrication value corresponding to a particle moving in a fluid with the effective viscosity given by the Einstein formula.     

Shape analysis based critical Eotvos numbers for buoyancy induced partial detachment of oil drops from hydrophilic surfaces

Removal of oil drops from solid surfaces immersed in an aqueous medium is of interest in many applications. It has been shown that drop shape analysis can be used to predict conditions at which the stability limit of a lighter than water oil drop on a solid surface immersed in an aqueous bath is reached (Adv. Colloid Interface Sci. 98 (2002) 265). However the above analysis is restricted to cases where the contact angle made by the drop is below 90degrees and when the surface conditions result in a 'pinned' contact line. In this paper, it is shown that drop shape analysis can be used to predict the critical conditions at which drop stability limit is reached for drop contact angles of 90degrees and above, which is encountered with 'hydrophilic' surfaces. This critical condition can predict the occurrence of partial oil drop detachment, before complete removal due to 'roll-up', which occurs when the hydrophilic surface is adequately smooth which prevents 'pinning' of the contact line. The critical conditions at which partial drop detachment occurs can also be approximately predicted from simple force balances. It has been shown (Adv. Colloid Interface Sci. 98 (2002) 265) that for contact angles less than 90degrees, the critical limit based on shape analysis appears to resolve the differences that arise due to alternate expressions for capillary retention force. This paper shows that even for contact angles above 90degrees, the critical conditions predicted from the shape analysis resolves the differences in the predictions from the alternate force balances. Drop shape analysis used in this paper is based on the 'Arc-length' form of Young-Laplace or 'drop shape' equation, which is different from the 'Y vs X' form of the above equation that is used in Adv. Colloid Interface Sci. 98 (2002) 265. The above drop shape equation is solved by a fourth order Runge-Kutta technique and it is shown that for angles less than 90degrees, the two forms of the drop shape equation, predict almost identical values of the critical Eotvos number. This paper highlights the competing effects of interfacial tension lowering induced drop instability and 'roll-up', a term that is used to describe the retraction of the contact line of an oil drop on a surface, in being the primary c ause for drop detachment.     

Buoyancy-driven convection around chemical fronts traveling in covered horizontal solution layers

Density differences across an autocatalytic chemical front traveling horizontally in covered thin layers of solution trigger hydrodynamic flows which can alter the concentration profile. We theoretically investigate the spatiotemporal evolution and asymptotic dynamics resulting from such an interplay between isothermal chemical reactions, diffusion, and buoyancy-driven convection. The studied model couples the reaction-diffusion-convection evolution equation for the concentration of an autocatalytic species to the incompressible Stokes equations ruling the evolution of the flow velocity in a two-dimensional geometry. The dimensionless parameter of the problem is a solutal Rayleigh number constructed upon the characteristic reaction-diffusion length scale. We show numerically that the asymptotic dynamics is one steady vortex surrounding, deforming, and accelerating the chemical front. This chemohydrodynamic structure propagating at a constant speed is quite different from the one obtained in the case of a pure hydrodynamic flow resulting from the contact between two solutions of different density or from the pure reaction-diffusion planar traveling front. The dynamics is symmetric with regard to the middle of the layer thickness for positive and negative Rayleigh numbers corresponding to products, respectively, lighter or heavier than the reactants. A parametric study shows that the intensity of the flow, the propagation speed, and the deformation of the front are increasing functions of the Rayleigh number and of the layer thickness. In particular, the asymptotic mixing length and reaction-diffusion-convection speed both scale as square root Ra for Ra>5. The velocity and concentration fields in the asymptotic dynamics are also found to exhibit self-similar properties with Ra. A comparison of the dynamics in the case of a monostable versus bistable kinetics is provided. Good agreement is obtained with experimental data on the speed of iodate-arsenous acid fronts propagating in horizontal capillaries. We furthermore compare the buoyancy-driven dynamics studied here to Marangoni-driven deformation of traveling chemical fronts in solution open to the air in the absence of gravity previously studied in the same geometry [L. Rongy and A. De Wit, J. Chem. Phys. 124, 164705 (2006)].     

Drop mixing in a microchannel for lab-on-a-chip platforms

We present theory, simulations, and experiments for discrete drop mixing in microchannels. The drops are placed sequentially in a channel and then moved at a set velocity to achieve mixing. The mixing occurs in three different regimes (diffusion-dominated, dispersion-dominated, and convection-dominated) depending on the Péclet number (Pe) and the drop dimensions. Introducing the modified Péclet number (Pe*), we show asymptotic curves that can be used to predict the mixing time and the required distance for mixing for any of the three regimes. Simulations of the mixing experiments using COMSOL agree with the theoretical limits. In our experimental work, we used a polydimethylsiloxane (PDMS) microchannel with a membrane air bypass valve to remove the air between drops. This approach enables precise control of the mixing and merging site. Experimental, simulation, and theoretical results all agree and show that mixing can occur in fractions of a second to hours, depending on the parameters used.     

Drop retention force as a function of drop size

The force, f, required to slide a drop past a surface is often considered in the literature as linear with the drop width, w, so that f/w = const. Furthermore, according to the Dussan equation for the case that the advancing and receding contact angles are constant with drop size, one can further simplify the above proportionality to f/V(1/3) = const where V is the drop volume. We show, however, that experimentally f/V(1/3) is usually a decaying function of V (rather than constant). The retention force increases with the time the drop rested on the surface prior to sliding. We show that this rested-time effect is similar for different drop sizes, and thus the change of f/V(1/3) with V occurs irrespective of the rested-time effect which suggests that the two effects are induced by different physical phenomena. The time effect is induced by the unsatisfied normal component of the Young equation which slowly deforms the surface with time, while the size effect is induced by time independent properties. According to the Dussan equation, the change of f/V(1/3) with V is also expressed in contact angle variation. Our results, however, show that contact angle variation that is within the scatter suffices to explain the significant force variation. Thus, it is easier to predict contact angle variation based on force variation rather than predicting force variation based on contact angle variation. A decrease of f/V(1/3) with V appears more common in the system studied compared to an increase.     

Effects of injection angle on the measurement of surface tension coefficient by drop weight method

The drop weight method for surface tension measurement is based on the weight of drops detached from a nozzle. The original idea was based on a postulate introduced by Tate (On the Magnitude of a Drop of Liquid Formed Under Different Circumstances, Philos. Mag. 27, 176–180, 1864), assuming that the weights of an ideal pendant drop and a detached ideal drop are identical, and that this weight is equal to the surface force that holds a drop attached to the nozzle. To consider the real volume of a drop that detaches from a nozzle, the method required a correction factor. Harkins and Brown suggested such correction factors for vertical injection from a nozzle. In this study, a correction factor for injection at different angles is presented and some of the hydrodynamic effects on surface tension measurement based on the drop weight method are studied. In addition, a model is introduced for the detachment time of drops in directions other than the vertical direction     

Degradation of polymers in high speed rotary homogenizers: A hydrodynamic interpretation

Dilute solutions of DNA in neutral phosphate buffer and of polystyrene in toluene are degraded to limiting molecular weights by high speed stirring in a Virtis homogenizer equipped with special thin, highly sharpened blades. Results are interpreted and justified in terms of the hydrodynamic theory of laminar boundary layers. Estimates of critical stress values for chain scission are obtained at various limiting molecular weights for the two polymer systems, and the treatment is used to rationalize the dependence of degradation upon homogenizer speed. The agreement between the hydrodynamic theory and experimental data implies that limiting molecular degradation occurs entirely in the boundary layer region extending a distance equal to approximately the molecular contour length from the leading edge of the blade tip and that laminar flow conditions prevail in this region. It therefore seems likely that polymer molecules subjected to a critical shearing stress are essentially completely extended in the streamlines of flow. The hydrodynamic theory is finally used to explain the peculiar kinetics of molecular degradation in high speed rotary homogenizers.     

Interaction of elastic bodies via surface forces. 2. Exponential decay

Our goal is to study theoretically the effect of deformation on the exponentially decaying interaction of two elastic solids separated by a thin liquid film. The deformed shape of the surfaces and the contribution of the elasticity to the total force, i.e., an additional term present between elastic bodies, are calculated from continuum elastic theory via a new asymptotic technique. Both the deformation and the contribution of the elasticity to the force are found to be significant on the length scale over which the surface force acts. The surface deformation is exponentially decaying with a decay length equal to that of the original surface interaction. It is especially important for large and/or rapidly changing force. The contribution of the elasticity is also exponentially decaying, but with half the decay length. Its strength depends on the elastic constants and size of the solids and on the magnitude and gradient of the original surface force. Depending on how the separation is detected, it can appear either as an attractive or as a repulsive contribution to the force. Our results open the possibility of recalculating the measured force to the interaction free energy.