Spectrum of position fluctuations of a Brownian particle bound in a harmonic trap near a plane wall

The spectrum of position fluctuations of a Brownian particle bound in a harmonic trap near a plane wall is calculated from an approximate result for the Fourier transform of the velocity autocorrelation function. Both a no-slip and a perfect slip boundary condition at the wall are considered. In both cases at low frequency the calculated spectrum differs markedly from recent experimental data. It is suggested that a partial slip boundary condition with a frequency-dependent slip coefficient may explain the experimental results.     

Effect of surfactant on retention behaviors of polystyrene latex particles in sedimentation field-flow fractionation: effective boundary slip model approach

A retention theory in sedimentation field-flow fractionation (SdFFF) was developed by exploiting the effective slip boundary condition (BC) that allows a finite velocity for particles to have at the wall, thereby alleviating the limitations set by the no-slip BC constraint bound to the standard retention theory (SRT). This led to an expression for the retention ratio R as R = (R(o) + v*(b))/(R(o) + v*(b)), where R(o) is the sterically corrected SRT retention ratio and v*(b) is the reduced boundary velocity. Then, v*(b) was modeled as v*(b) = v*(b,o)/[1 + (7K*S(o))(1/2)], where S(o) is the surfactant (FL-70) concentration and K* is the distribution coefficient associated with the langmuirian isotherm of the apparent effective mass against S(o). We applied this to study the surfactant effect on the retention behaviors of polystyrene (PS) latex beads of 170-500 nm in diameter. As a result, an empirical relation was found to hold between v*(b,o) and d(o) (estimated from R(o) at S(o) = 0) as v*(b,o) - v*(o,o)[1 - (d(c)/d(o))], where v*(o,o) is the asymptotic value of v*(b,o) in the vanishing d(c)/d(o) limit and d(c) is the cutoff value at which v*(b,o) would vanish. According to the present approach, the no-slip BC (v*(b,o) = 0) was predicted to recover when d(o) ～ d(c), and the boundary slip effect could be significant for S(o) ≤ 0.05%, particularly for large latex beads.     

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.     

Nanohydrodynamics: the intrinsic flow boundary condition on smooth surfaces

A dynamic surface force apparatus is used to determine the intrinsic flow boundary condition of two simple liquids, water and dodecane, on various smooth surfaces. We demonstrate the impact of experimental errors and data analysis on the accuracy of slip length determination. In all systems investigated, the dissipation is described by a well-defined boundary condition accounting for a whole range of separation, film thickness, and shear rate. A no-slip boundary condition is found in all wetting situations. On strongly hydrophobic surfaces, water undergoes finite slippage that increases with hydrophobicity. We also compare the relative influence of hydrophobicity and liquid viscosity on boundary flow by using water-glycerol mixtures with similar wetting properties.     

Contact line motion in confined liquid-gas systems: Slip versus phase transition

In two-phase flows, the interface intervening between the two fluid phases intersects the solid wall at the contact line. A classical problem in continuum fluid mechanics is the incompatibility between the moving contact line and the no-slip boundary condition, as the latter leads to a nonintegrable stress singularity. Recently, various diffuse-interface models have been proposed to explain the contact line motion using mechanisms missing from the sharp-interface treatments in fluid mechanics. In one-component two-phase (liquid-gas) systems, the contact line can move through the mass transport across the interface while in two-component (binary) fluids, the contact line can move through diffusive transport across the interface. While these mechanisms alone suffice to remove the stress singularity, the role of fluid slip at solid surface needs to be taken into account as well. In this paper, we apply the diffuse-interface modeling to the study of contact line motion in one-component liquid-gas systems, with the fluid slip fully taken into account. The dynamic van der Waals theory has been presented for one-component fluids, capable of describing the two-phase hydrodynamics involving the liquid-gas transition [A. Onuki, Phys. Rev. E 75, 036304 (2007)]. This theory assumes the local equilibrium condition at the solid surface for density and also the no-slip boundary condition for velocity. We use its hydrodynamic equations to describe the continuum hydrodynamics in the bulk region and derive the more general boundary conditions by introducing additional dissipative processes at the fluid-solid interface. The positive definiteness of entropy production rate is the guiding principle of our derivation. Numerical simulations based on a finite-difference algorithm have been carried out to investigate the dynamic effects of the newly derived boundary conditions, showing that the contact line can move through both phase transition and slip, with their relative contributions determined by a competition between the two coexisting mechanisms in terms of entropy production. At temperatures very close to the critical temperature, the phase transition is the dominant mechanism, for the liquid-gas interface is wide and the density ratio is close to 1. At low temperatures, the slip effect shows up as the slip length is gradually increased. The observed competition can be interpreted by the Onsager principle of minimum entropy production.     

Flow boundary conditions for chain-end adsorbing polymer blends

Using the phenol-terminated polycarbonate blend as an example, we demonstrate that the hydrodynamic boundary conditions for a flow of an adsorbing polymer melt are extremely sensitive to the structure of the epitaxial layer. Under shear, the adsorbed parts (chain ends) of the polymer melt move along the equipotential lines of the surface potential whereas the adsorbed additives serve as the surface defects. In response to the increase of the number of the adsorbed additives the surface layer becomes thinner and solidifies. This results in a gradual transition from the slip to the no-slip boundary condition for the melt flow, with a nonmonotonic dependence of the slip length on the surface concentration of the adsorbed ends.     

Drop impact on surfactant films and solutions

Drops impacting on horizontal aqueous surfactant films have been analyzed using a high-speed camera. Drops of either water or aqueous surfactant solutions had a diameter of 2.4?±?0.4 mm and impacted with a velocity of 0.1 to 1.3 m/s. As surfactants, anionic sodium dodecyl sulfate and cationic cetyltrimethyl ammonium bromide were used. Pure water drops impacting on freestanding surfactant films showed coalescence, bouncing, partial bouncing, passing, and partial passing. For bouncing, the concentration of surfactant in the surfactant film must exceed the critical micelle concentration. When surfactant was added to the drop, coalescence and partial passing were suppressed. We attribute the different behavior to different hydrodynamic boundary conditions at the surface of pure water and surfactant solution, leading to different repulsive hydrodynamic forces arising when the air has to flow out of the closing gap between the two liquid surfaces. The boundary condition changes as a function of surfactant concentration from a slip to no-slip, leading to stronger hydrodynamic repulsion. In addition, estimates of the characteristic velocities show that diffusion of air into the water is slow and can only account for the very last thinning of the air gap before coalescence.     

Steady-state hydrodynamics of a viscous incompressible fluid with spinning particles

The steady-state hydrodynamics of a viscous incompressible fluid with spinning particles is studied on the basis of extended Stokes equations. The profiles of flow velocity and spin velocity in simple flow situations may be used to determine the vortex viscosity and spin viscosity of the molecular liquid or fluid suspension. As an example, one situation studied is the flow generated by a uniform torque density in a planar layer of infinite fluid. The spinning particles drive a nearly uniform flow on either side of the layer, in opposite directions on the two sides. The Green function of the extended Stokes equations is derived. The translational and rotational friction coefficients of a sphere with no-slip boundary conditions, and the corresponding flow profiles, are calculated.     

Passive solute separation in AC electroosmosis including surface charge-coupled hydrodynamic slip effects

Separation of electrically neutral, mutually noninteracting passive solutes via AC electroosmotic slit channel flows is investigated for general asymmetric wall surface zeta potentials and apparent hydrodynamic slip lengths. We consider the nontrivial coupling between the surface potentials (or charge densities) and the apparent slip lengths, and focus our attention on the occurrence of a so called “crossover phenomenon” for separating out the slow diffusers when both slow and fast diffusers are present. Results show that regardless of the potential-slip coupling, wider bandwidths become available for crossover phenomenon to occur when the electroosmotic velocity gradient (magnitude) is greater. Contrarily, plug-like velocity profiles inhibit crossover phenomenon, and the potential-slip parametric combinations leading to such profiles can be easily identified by the conditions for minimal transport enhancement reported in recent literature. When separating out the slow diffuser or crossover phenomenon is desired, we recommend incorporating significant asymmetry in the surface potential and apparent slip boundary conditions such that the operating frequency and flow oscillation amplitude may be lowered to more practical values. Our results also agree with and strengthen the physical picture for explaining crossover phenomenon in macroscopic pressure-driven oscillatory flows.     

Spreading of fluids on solids under pressure: slip and stick effects

Spreading of different types of fluid on solids under an impressed force is an interesting problem. Here we study spreading of four fluids, having different hydrophilicity and viscosity on two substrates - glass and perspex, under an external force. The area of contact of fluid and solid is video-photographed and its increase with time is measured. The results for different external forces can be scaled onto a common curve. We try to explain the nature of this curve on the basis of existing theoretical treatment where either the no-slip condition is used or slip between fluid and substrate is introduced. We find that of the eight cases under study, in five cases quantitative agreement is obtained using a positive slip coefficient. The remaining three can be explained with a negative slip coefficient, equivalent to a sticking effect.     

Numerical Analysis of 3‐D Flow Past a Stationary Sphere with Slip Condition at Low and Moderate Reynolds Numbers

Computations are performed to determine the steady 3‐D viscous fluid flow forces acting on the stationary spherical suspended particle at low and moderate Reynolds numbers in the range of 0.1≤Re≤200. A slip is supposed on the boundary so that the slip velocity becomes proportional to the shear stress. This model possesses a single parameter to account for the slip coefficient λ (Pa.s/m), which is made dimensionless and is called Trostel number (Tr=λ a/μ). Decreasing slip, increases drag in all Reynolds limits, but slip has smaller effects on drag coefficient at lower Reynolds number regimes. Increasing slip at known Reynolds number causes to delay of flow separation and inflect point creation in velocity profiles. At full slip conditions, shear drag coefficient will be zero and radial drag coefficient reaches to its maximum values. Flow around of sphere at full‐slip condition is not equal to potential flow around a sphere. Present numerical results corresponding to full slip (Tr→0) are in complete accord with certain results of flow around of inviscid bubbles, and the results corresponding to no‐slip (Tr→∞) have excellent agreement with the results predicted by the no‐slip boundary condition.     

General continuum boundary conditions for miscible binary fluids from molecular dynamics simulations

Molecular dynamics simulations are used to explore the flow behavior and diffusion of miscible fluids near solid surfaces. The solid produces deviations from bulk fluid behavior that decay over a distance of the order of the fluid correlation length. Atomistic results are mapped onto two types of continuum model: Mesoscopic models that follow this decay and conventional sharp interface boundary conditions for the stress and velocity. The atomistic results, and mesoscopic models derived from them, are consistent with the conventional Marangoni stress boundary condition. However, there are deviations from the conventional Navier boundary condition that states that the slip velocity between wall and fluid is proportional to the strain rate. A general slip boundary condition is derived from the mesoscopic model that contains additional terms associated with the Marangoni stress and diffusion, and is shown to describe the atomistic simulations. The additional terms lead to strong flows when there is a concentration gradient. The potential for using this effect to make a nanomotor or pump is evaluated.     

Broadside mobility of a disk in a viscous fluid near a plane wall with no-slip boundary condition

The broadside motion of a disk in a viscous fluid towards a planar wall with no-slip boundary condition is studied on the basis of the steady-state Stokes equations. It is shown that flow velocity and pressure of the fluid can be found conveniently from a superposition of elementary complex stream functions. The two amplitude functions characterizing the superposition are found from the numerical solution of a pair of integral equations for the axial and radial velocity components at the disk. The numerical procedure converges fast, provided the distance to the plane is not much smaller than the radius of the disk. For small distance the flow is well approximated by lubrication theory.     

Slip length of water on graphene: limitations of non-equilibrium molecular dynamics simulations

Data for the flow rate of water in carbon nanopores is widely scattered, both in experiments and simulations. In this work, we aim at precisely quantifying the characteristic large slip length and flow rate of water flowing in a planar graphene nanochannel. First, we quantify the slip length using the intrinsic interfacial friction coefficient between water and graphene, which is found from equilibrium molecular dynamics (EMD) simulations. We then calculate the flow rate and the slip length from the streaming velocity profiles obtained using non-equilibrium molecular dynamics (NEMD) simulations and compare with the predictions from the EMD simulations. The slip length calculated from NEMD simulations is found to be extremely sensitive to the curvature of the velocity profile and it possesses large statistical errors. We therefore pose the question: Can a micrometer range slip length be reliably determined using velocity profiles obtained from NEMD simulations? Our answer is "not practical, if not impossible" based on the analysis given as the results. In the case of high slip systems such as water in carbon nanochannels, the EMD method results are more reliable, accurate, and computationally more efficient compared to the direct NEMD method for predicting the nanofluidic flow rate and hydrodynamic boundary condition.     

Slip flow in graphene nanochannels

We investigate the hydrodynamic boundary condition for simple nanofluidic systems such as argon and methane flowing in graphene nanochannels using equilibrium molecular dynamics simulations (EMD) in conjunction with our recently proposed method [J. S. Hansen, B. D. Todd, and P. J. Daivis, Phys. Rev. E 84, 016313 (2011)]. We first calculate the fluid-graphene interfacial friction coefficient, from which we can predict the slip length and the average velocity of the first fluid layer close to the wall (referred to as the slip velocity). Using direct nonequilibrium molecular dynamics simulations (NEMD) we then calculate the slip length and slip velocity from the streaming velocity profiles in Poiseuille and Couette flows. The slip lengths and slip velocities from the NEMD simulations are found to be in excellent agreement with our EMD predictions. Our EMD method therefore enables one to directly calculate this intrinsic friction coefficient between fluid and solid and the slip length for a given fluid and solid, which is otherwise tedious to calculate using direct NEMD simulations at low pressure gradients or shear rates. The advantages of the EMD method over the NEMD method to calculate the slip lengths/flow rates for nanofluidic systems are discussed, and we finally examine the dynamic behaviour of slip due to an externally applied field and shear rate.     

Structure and rheological behavior of highly charged colloidal particles in a cylindrical pore I. Effect of pore size

In this work we performed nonequilibrium Brownian dynamics (NEBD) computer simulations of highly charged colloidal particles in diluted suspension under a parabolic flow in cylindrical pores. The influence of charged and neutral cylindrical pores on the structure and rheology of suspensions is analyzed. A shear-induced disorder-order-disorder-like transition was monitored for low shear rates and small pore diameters. We calculate the concentration profiles, axial distribution functions, and axial-angular pair correlation functions to determine the structural properties at steady state for a constant shear flow for different pore sizes and flow strengths. Similar behavior has been observed in a planar narrow channel in the case of charged interacting colloidal particles (M.A. Valdez, O. Manero, J. Colloid Interface Sci. 190 (1997) 81). The mobility of the particles in the radial direction decreases rapidly with the flow and becomes practically frozen. The flow exhibits non-Newtonian shear thinning behavior due to interparticle interactions and particle-wall interaction; the apparent viscosity is lower as the pore diameter decreases, giving rise to an apparent slip in the colloidal suspension. The calculated slip velocity was higher than that obtained in a rectangular slit under shear flow.     

Mobility matrix of a spherical particle translating and rotating in a viscous fluid confined in a spherical cell, and the rate of escape from the cell

The mobility matrix of a spherical particle moving in a spherical cavity, filled with a viscous incompressible fluid, and with no-slip boundary condition at the wall of the cavity, is evaluated from the Oseen tensor for the cavity by the method used by Lorentz for a particle near a planar wall. For the case that the particle is a rigid sphere with no-slip boundary condition the comparison with exact calculations shows that the approximation is quite accurate, provided the radius of the particle is small relative to that of the cavity, and provided the particle is not too close to the wall. The translational mobility is used to derive the diffusion tensor of a Brownian particle via an Einstein relation. The approximate result for the diffusion tensor is employed to estimate the rate of escape of a Brownian particle from a cavity with semipermeable wall.     

Thermophoretic Motion of a Sphere Parallel to an Insulated Plane

An analytical study is presented for the thermophoresis of a sphere in a constant applied temperature gradient parallel to an adiabatic plane. The Knudsen number is assumed to be small so that the fluid flow can be described by a continuum model with a thermal creep and a hydrodynamic slip at the particle surface. A method of reflections is used to obtain the asymptotic formulas for the temperature and velocity fields in the quasisteady situation. The thermal insulated plane may be a solid wall (no-slip) and/or a free surface (perfect-slip). The boundary effect on the thermophoretic motion is found to be weaker than that on the axisymmetric thermophoresis of a sphere normal to a plane with constant temperature. In comparison with the motion driven by gravitational force, the interaction between the particle and the boundary is less significant under thermophoresis. Even so, the interaction between the plane and the particle can be very strong when the gap thickness approaches zero. For the thermophoretic motion of a particle parallel to a solid plane, the effect of the plane surface is to reduce the translational velocity of the particle. In the case of particle migration parallel to a free surface due to thermophoresis, the translating velocity of a particle can be either greater or smaller than that which would exist in the absence of the plane surface, depending on the relative thermal conductivity and the surface properties of the particle and its relative distance from the plane. Not only the translational velocity but also the rotational velocity of the thermophoretic sphere near the plane boundary is formulated analytically. The rotating direction of the particle is strongly dominated by its surface properties and the internal-to-external thermal conductivity. Besides the particle motion, the thickness of the thermophoretic boundary layer is evaluated by considering the thermophoretic mobility. Generally speaking, a free surface exerts less influence on the particle movement than a solid wall. Copyright 2000 Academic Press.     

The effect of rotational and translational energy exchange on tracer diffusion in rough hard sphere fluids

A study is presented of tracer diffusion in a rough hard sphere fluid. Unlike smooth hard spheres, collisions between rough hard spheres can exchange rotational and translational energy and momentum. It is expected that as tracer particles become larger, their diffusion constants will tend toward the Stokes-Einstein hydrodynamic result. It has already been shown that in this limit, smooth hard spheres adopt "slip" boundary conditions. The current results show that rough hard spheres adopt boundary conditions proportional to the degree of translational-rotational energy exchange. Spheres for which this exchange is the largest adopt "stick" boundary conditions while those with more intermediate exchange adopt values between the "slip" and "stick" limits. This dependence is found to be almost linear. As well, changes in the diffusion constants as a function of this exchange are examined and it is found that the dependence is stronger than that suggested by the low-density, Boltzmann result. Compared with smooth hard spheres, real molecules undergo inelastic collisions and have attractive wells. Rough hard spheres model the effect of inelasticity and show that even without the presence of attractive forces, the boundary conditions for large particles can deviate from "slip" and approach "stick."