Stick Boundary Conditions Research Articles

Exploration of Stokes hydrodynamic law at molecular length scales.

The celebrated generalized Stokes law predicts that the velocity of a particle pulled through a liquid by an external force, Fex, is directly proportional to the force and inversely proportional to the friction ζ acted by the medium on the particle. We investigate the range of validity of the generalized Stokes law at molecular length scales by employing computer simulations to calculate friction by pulling a tagged particle with a constant force. We thus calculate friction for two model interaction potentials, Lennard-Jones and soft sphere, for several particle sizes, ranging from radius (a) smaller than the solvent particles to three times larger. We next obtain friction from diffusion (D) by using Einstein's relation between diffusion and friction ζ in an unperturbed liquid. We find a quantitative agreement between the two at a small-to-intermediate pulling force regime for all the sizes studied. The law does break down at a large pulling force beyond a threshold value. Importantly, the range of validity of Stokes' scheme to obtain friction increases substantially if we turn off the attractive part of the interaction potential. Additionally, we calculate the viscosity (η) of the unperturbed liquid and find a good agreement with the Stokes-Einstein relation ζ = Cηa for the viscosity dependence with a value of C close to 5 π, which is intermediate between the slip and stick boundary condition.

Experimental and Theoretical Rotational Diffusion Studies of 7DM4M1M1,8, N-2(1H)-one and 7A4T2H1B-2-one in Series of Alcohol Solvents: Stoke's-Einstein-Debye and Alavi-Waldeck Models.

Rotational diffusion studies of two solutes 7-(dimethylamino)-4-methoxy-1-methyl-1,8-naphthyridin-2(1H)-one (7DM4M1M1,8, N-2(1H)-one) and 7-amino-4-(trifluoromethyl)-2H-1-benzopyran-2-one (7A4T2H1B-2-one) having equal volumes but different chemical natures are studied in series of alcohol solvents at 303K using steady-state methods. HOMO-LUMO, Electron density, Molecular electrostatic potential (MEP), etc., are obtained from computational calculations using Gaussian 09 software. Rotational reorientation times of 7DM4M1M1,8, N-2(1H)-one solute molecule is found to be less than 7A4T2H1B-2-one solute molecule indicates it rotates slowly in chosen solvents. Stoke's-Einstein-Debye (SED) model with stick boundary conditions for the 7A4T2H1B-2-one solute molecule is modeled to describe mechanical friction. Polar solutes along with mechanical friction also experience dielectric friction. Both these frictions being non-separable, the Alavi-Waldeck (AW) model is studied for dielectric friction contribution to the total friction solute experiences in solvents. AW model effectively explains the observed dielectric friction in alcohol solvents.

Drag on nanoparticles in a liquid: from slip to stick boundary conditions.

Stokes' law with stick boundary conditions has been widely accepted for the transport of microscale particles in a liquid. For nanoparticles, however, the hydrodynamic boundary conditions become unclear. In this work, the drag force acting on nanoparticles suspended in a liquid and the hydrodynamic boundary coefficient were calculated by using molecular dynamics simulations. For weak interfacial couplings, slip boundary conditions can be used to describe the particle transport, whereas at strong interfacial couplings, the hydrodynamic boundary coefficient converges to a value greater than the prediction by Stokes' law. In the present paper, we propose a density accumulation length to determine the effective particle size, which makes Stokes' law valid for nanoparticles. For a copper nanoparticle suspended in an argon liquid, the density accumulation length increases to 0.32 nm with increasing solid-liquid coupling strength. Furthermore, it is found that there exists a transition from slip to stick boundary conditions as the solid-liquid intermolecular coupling strength increases. The results presented in this work provide guidance for the prediction and manipulation of the transport properties of nanoparticles in a liquid.

Solute dynamics of a hydrophobic molecule in a menthol-thymol based type-V deep eutectic solvent: effect of composition of the components.

Type-V deep eutectic solvents (DESs) are a newly emerging unique class of solvents obtained by physical mixing and heating of non-ionic components. These solvents show deviation from the thermodynamic ideality. Compared to type-I to IV DESs, type-V DESs are less explored and their physical chemistry is in its nascent stage. In this work, we have chosen a type-V DES based on menthol-thymol (MT) for our working media. Solvent and rotational dynamics were studied with varying temperature using a well-known solvatochromic probe, Coumarin 153 (C153). We prepared the MT-based DES using a reported procedure at three molar ratios: 1 : 1 (M1T1), 1 : 1.5 (M1T1.5), and 2 : 1 (M2T1) of menthol (M) and thymol (T). Time-resolved emission spectra (TRES) were constructed with varying temperature. Utilizing TRES, the decay of the solvent correlation function (C(t)) was plotted. We have correlated the solvent relaxation time in these DESs as a function of viscosity. The time-resolved anisotropy decays were also collected to perceive the rotational relaxation dynamics of C153 as a function of temperature. The decay of solvent relaxation was found to be bi-exponential, and the average solvation time (〈τs〉) in M2T1 was found to be longer than those of M1T1.5 and M1T1. The rotational reorientation times (〈τrot〉) also follow the same trend. We have analysed the rotational dynamics of C153 in type-V DESs employing the Stokes-Einstein-Debye (SED) hydrodynamic model. The rotational dynamics in DESs demonstrate a good correlation with the SED model with a little deviation. In MT-based DESs, the solute's rotational relaxation times approach hydrodynamic stick boundary condition at low viscosity (or at high temperatures) for all molar compositions. Using the Arrhenius-type equations, we have correlated the activation energies for the rotational motion of C153, along with the viscous flow and non-radiative pathways for all the DESs.

Kelvin-Voigt equations with memory: existence, uniqueness and regularity of solutions

In general, the study of inverse problems is realizable only in the case when the corresponding direct problems have the unique solution with some necessary properties such as continuity and regularity. In this paper, we study initial-boundary value problems for the system of 2D-3D nonlinear Kelvin-Voigt equations with memory, which describes a motion of an incompressible homogeneous non-Newtonian fluids with viscoelastic and relaxation properties. The investigation of these direct problems is related to the study of inverse problems for this system, which requires the continuity and regularity of solutions to these direct problems and their derivatives. The system, in addition to the initial condition, is supplemented with one of the boundary conditions: stick and slip boundary conditions. In both cases of these boundary conditions, the global in time existence and uniqueness of strong solutions to these initial-boundary value problems were proved. Moreover, under suitable assumptions on the data, the regularity of solutions and their derivatives were established.

Inverse problems for nonlinear Navier–Stokes–Voigt system with memory

This paper deals with the unique solvability of some inverse problems for nonlinear Navier–Stokes–Voigt (Kelvin–Voigt) system with memory that governs the flow of incompressible viscoelastic non-Newtonian fluids. The inverse problems that study here, consist of determining a time dependent intensity of the density of external forces, along with a velocity and a pressure of fluids. As an additional information, two types of integral overdetermination conditions over space domain are considered. The system supplemented also with an initial and one of the boundary conditions: stick and slip boundary conditions. For all inverse problems, under suitable assumptions on the data, the global and local in time existence and uniqueness of weak and strong solutions were established.

Analysing Sources of Error in Total Internal Reflection Microscopy (TIRM) Experiments and Data Analysis.

Many phenomena observed in synthetic and biological colloidal suspensions are dominated by the static interaction energies and the hydrodynamic interactions that act both between individual particles and also between colloids and macroscopic interfaces. This calls for methods that allow precise measurements of the corresponding forces. One method used for this purpose is total internal reflection microscopy (TIRM), which has been employed for around three decades to measure in particular the interactions between a single particle suspended in a liquid and a solid surface. However, given the importance of the observable variables, it is crucial to understand the possibilities and limitations of the method. In this paper, we investigate the influence of technically unavoidable noise effects and an inappropriate choice of particle size and sampling time on TIRM measurement results. Our main focus is on the measurement of diffusion coefficients and drift velocities, as the influence of error sources on dynamic properties has not been investigated so far. We find that detector shot noise and prolonged sampling times may cause erroneous results in the steep parts of the interaction potential where forces of the order of pico-Newtons or larger act on the particle, while the effect of background noise is negligible below certain thresholds. Furthermore, noise does not significantly affect dynamic data but we find that lengthy sampling times and/or probe particles with too small a radius will cause issues. Most importantly, we observe that dynamic results are very likely to differ from the standard hydrodynamic predictions for stick boundary conditions due to partial slip.

Self-diffusion coefficients of heavy water in low n-alkanes

The self-diffusion coefficients of D2O (i.e., heavy water or 2H2O) in eight n-alkanes at 298 K were investigated using 2H NMR diffusometry. D tended to decrease in the n-alkanes with increasing carbon number and be inversely proportional to the n-alkane viscosity to a fractional power. The hydrodynamic radius of D2O in the n-alkanes, calculated from Stokes-Einstein-Sutherland equation with stick boundary conditions, decreased with increasing n-alkane size and was smaller in the n-alkanes than in the bulk phase due to the absence of hydrogen bonds. This study provides fundamental data on the diffusion properties of water molecules dissolved in n-alkanes with applications to, for instance, molecular dynamics simulations or interpretation of Ostwald ripening.

Ice friction at the nanoscale

The origin of ice slipperiness has been a matter of great controversy for more than a century, but an atomistic understanding of ice friction is still lacking. Here, we perform computer simulations of an atomically smooth substrate sliding on ice. In a large temperature range between 230 and 266 K, hydrophobic sliders exhibit a premelting layer similar to that found at the ice/air interface. On the contrary, hydrophilic sliders show larger premelting and a strong increase of the first adsorption layer. The nonequilibrium simulations show that premelting films of barely one-nanometer thickness are sufficient to provide a lubricating quasi-liquid layer with rheological properties similar to bulk undercooled water. Upon shearing, the films display a pattern consistent with lubricating Couette flow, but the boundary conditions at the wall vary strongly with the substrate's interactions. Hydrophobic walls exhibit large slip, while hydrophilic walls obey stick boundary conditions with small negative slip. By compressing ice above atmospheric pressure, the lubricating layer grows continuously, and the rheological properties approach bulk-like behavior. Below 260 K, the equilibrium premelting films decrease significantly. However, a very large slip persists on the hydrophobic walls, while the increased friction on hydrophilic walls is sufficient to melt ice and create a lubrication layer in a few nanoseconds. Our results show that the atomic-scale frictional behavior of ice is a combination of spontaneous premelting, pressure melting, and frictional heating.

Validity of the Stokes−Einstein relation in liquid 3d transition metals for a wide range of temperatures

The validity of Stokes−Einstein (SE) relations obtained from both slip and stick boundary conditions has been investigated for an extending temperature range approximately 250 K from the melting points of liquid Fe, Co and Ni using Dzugutov scaling scheme for transport coefficients. The basic ingredients of Dzugutov scaling scheme are the temperature dependent effective hard-sphere (HS) diameter, σ(T), and the excess entropy, Sex(T). Sex is calculated considering both two (S2) and many body approximations (ST) of liquid atoms. For both ingredients, we have applied variational modified hypernetted chain (VMHNC) integral equation theory in conjunction with both Brettonet−Silbert (BS) pseudopotential and many body potential obtained from embedded atom method (EAM). Calculated transport coefficients using the combination of EAM and ST are in better agreement with available literature values than any other applied combinations. We observe that SE relation holds for the studied representative systems when the combination of EAM potential and ST has been considered.

Drag on Sticky and Janus (Slip-Stick) Spheres Confined in a Channel

Abstract Flow over a sphere is a frequently occurring phenomenon in a range of industries. The sphere is confined in a channel in most of these applications. Therefore, it is important to understand the effect of confinement on the hydrodynamics of the flow around a sphere placed in a channel. These spheres can be bubbles, solid particles or droplets resulting in different boundary conditions (stick or slip) on the surface of the sphere. In recent years, Janus spheres having slip and stick boundary conditions on parts of the sphere have gained importance because of their potential applications. In this article, drag coefficient for a spherical particle fixed at the centerline of a channel of square cross-section is obtained computationally for stick, slip, and stick-slip surfaces of the sphere for a range of particle Reynolds numbers (1–80) and particle to channel size ratios (0.05–0.80). Further, the position of stick particle in the channel is varied to understand the effect of particle location on the drag coefficient. Correlations are proposed to calculate the drag coefficient for no-slip and Janus particles when the particle is at the channel center.

Numerical modelling and comparison of the temporal evolution of mantle and tails surrounding rigid elliptical objects in simple shear regime under stick and slip boundary conditions

Structures associated with rigid inclusions are a rich source of evidence to understand the local deformation regime. The behaviour of rigid objects in modelled here as being immersed in a linear Newtonian fluid with either (i) a stick boundary condition (continuity of stress and velocity across the boundary) or (ii) a slip boundary condition (continuity of boundary normal stress and velocity across the boundary with zero shear stress at the boundary). Of particular interest are the types of structures developed in a concentric region adjacent to the object termed the mantle. A model of the displacement of points around the inclusion comprises a set of ordinary differential equations which are solved numerically. A comprehensive set of simulations for a variety of mantle sizes, object aspect ratios, initial orientations as well as different boundary conditions has been performed. A comparison between natural examples and model output indicates a level of consistency. The resulting structures differ in detail and in a broader sense. In general δ-type structures only develop when stick boundary conditions are in operation. In contrast, σ-type structures at high strain are restricted to slip boundary conditions. Slip conditions also tend to be the source of complex mantle types involving more than one generation of mantle structures or wings. Furthermore, our model indicates that using asymmetry of orientation of objects relative to the shear direction may be problematic when used alone, particularly if stick boundary conditions prevail but that together with mantle structures there is less chance of confusion.

Translational and Rotational Diffusion Coefficients of Gold Nanorods Dispersed in Mixtures of Water and Glycerol by Polarized Dynamic Light Scattering.

Polarized dynamic light scattering (DLS) gives access to orientation-averaged translational and rotational diffusion coefficients of anisotropic particles dispersed in fluids in a single experiment. As the combination of both diffusivities contains information on the morphology of the particles, their simultaneous and accurate measurement for the same sample and thermodynamic state is beneficial for particle characterization. For nontransparent model suspensions of gold nanorods in water and water-glycerol mixtures, a scattering geometry in reflection direction was realized, which minimizes multiple scattering and allows using low laser powers to avoid laser heating. Furthermore, a heterodyne detection scheme was guaranteed by superimposing much stronger reference light to the scattered light. This ensures an unambiguous data evaluation and reduces the uncertainties for the rotational and the translational diffusivity, where the latter is accessible with smaller uncertainty. For the water-based suspensions, both diffusivities agree well with the stick hydrodynamic theory for rods and show an Andrade-type behavior in the studied temperature range from 271 to 323 K. The measured results for both diffusivities, particularly for the rotational diffusivity, indicate a breakdown of the stick boundary conditions for dynamic viscosities larger than 4 mPa·s.

Lattice Boltzmann electrokinetics simulation of nanocapacitors.

We propose a method to model metallic surfaces in Lattice Boltzmann Electrokinetics (LBE) simulations, a lattice-based algorithm rooted in kinetic theory which captures the coupled solvent and ion dynamics in electrolyte solutions. This is achieved by a simple rule to impose electrostatic boundary conditions in a consistent way with the location of the hydrodynamic interface for stick boundary conditions. The proposed method also provides the local charge induced on the electrode by the instantaneous distribution of ions under voltage. We validate it in the low voltage regime by comparison with analytical results in two model nanocapacitors: parallel plates and coaxial electrodes. We examine the steady-state ionic concentrations and electric potential profiles (and corresponding capacitance), the time-dependent response of the charge on the electrodes, and the steady-state electro-osmotic profiles in the presence of an additional, tangential electric field. The LBE method further provides the time-dependence of these quantities, as illustrated on the electro-osmotic response. While we do not consider this case in the present work, which focuses on the validation of the method, the latter readily applies to large voltages between the electrodes, as well as to time-dependent voltages. This work opens the way to the LBE simulation of more complex systems involving electrodes and metallic surfaces, such as sensing devices based on nanofluidic channels and nanotubes, or porous electrodes.

Near wall velocimetry on a rheometer

We present the principle and implementation of a novel in situ dynamic light scattering velocimetry on a commercial rheometer. The technique is based on total internal reflection at the bottom plate of a rheometer. The evanescent wave that penetrates a sub micrometer level into the sample enables the monitoring of the near wall (NW) dynamics under rheometric flow. Slip during shear is detected by measuring the NW velocity profile. We discuss how to extract the NW velocities from the measured autocorrelation functions, using Fourier transform analysis. The method was tested on a Newtonian fluid, colloidal dispersions, and polymer solutions. Stick boundary conditions were observed in the Newtonian fluid; the presence of strong slip was easily identified in viscoelastic colloidal suspensions; and polymer solutions were found to present weak slip. This high resolution NW velocimetry rheo-optical technique should prove valuable for a broad range of flowing materials.

Solvation and Rotational Diffusion of Solutes in Room Temperature Ionic Liquids as Studied by EPR Spectroscopy with Nitroxide Spin Probing Method

In this article, we briefly introduced our studies on solvation and rotational diffusion of solutes in room temperature ionic liquids (RTILs) by electron paramagnetic resonance with nitroxide spin probing method. Most of the rotational correlation times for the nitroxide radicals are within the range calculated on the basis of Stokes–Einstein–Debye hydrodynamic theory with stick and slip boundary conditions or Gierer–Wirtz theory except for smaller solutes in some RTILs with smaller BF4 and PF6 anions. In RTILs with 1-butyl-3-methylimidazolium as cation and BF4 or PF6 as anion, nitroxide radicals undergo rotational diffusion like supercooled liquids and nitroxide radical with smaller volume rotationally slips.

Stick boundary condition at large hard sphere arising from effective attraction in binary hard-sphere mixtures.

We have studied the diffusion of a large hard-sphere solute immersed in binary hard-sphere mixtures. We reveal how the boundary condition at the solute surface is affected by the solvent density around the solute. Solving equations for a binary compressible mixture by perturbation expansions, we obtain the boundary condition depending on the size ratio of binary solvent spheres. When the size ratio is 1:2, the boundary condition lies close to the slip boundary condition. By contrast, when the size ratio becomes large, the boundary condition approaches the stick boundary condition with the addition of larger solvent spheres. We find that the transition to the stick boundary condition is caused by the increase in the solvent density around the solute due to an entropic effect.

Is Solute Rotation in an Ionic Liquid Influenced by the Addition of Glucose?

Fluorescence anisotropy measurements and molecular dynamics (MD) simulations have been performed to understand the specific interactions of two structurally similar nondipolar solutes, 2,5-dimethyl-1,4-dioxo-3,6-diphenylpyrrolo[3,4-c]pyrrole (DMDPP) and 1,4-dioxo-3,6-diphenylpyrrolo[3,4-c]pyrrole (DPP), with neat 1-butyl-3-methylimidazolium dicyanamide ([BMIM][N(CN)2]) and also in the presence of glucose. It has been observed that the measured reorientation times of DMDPP in neat [BMIM][N(CN)2] follow the predictions of the Stokes-Einstein-Debye hydrodynamic theory with slip boundary condition. Addition of glucose (0.075 and 0.15 mole fraction) has no bearing on the rotational diffusion of the solute apart from the viscosity related effects. In contrast, the reorientation times of DPP in neat [BMIM][N(CN)2] obey stick boundary condition as the hydrogen bond donating solute experiences specific interactions with the dicyanamide anion. No influence of the additive can be noticed on the rotational diffusion of DPP at 0.075 mole fraction of glucose. However, at 0.15 mole fraction of glucose, the reorientation times of the solute at a given viscosity and temperature decrease by 15-40% compared to those obtained in the neat ionic liquid. MD simulations indicate that each DPP molecule hydrogen bonds with two dicyanamide anions in neat ionic liquid. The simulations also reveal that, at 0.15 mole fraction of glucose, the concentration of anions hydrogen bonded to glucose increases significantly; therefore, the percentage of solute molecules that can form hydrogen bonds with two dicyanamide anions decreases to 84, which leads to faster rotation of DPP.

Apparent slip of shear thinning fluid in a microchannel with a superhydrophobic wall.

The peculiarities of simple shear flow of shear thinning fluids over a superhydrophobic wall consisting of a set of parallel gas-filled grooves and solid stripes (domains with slip and stick boundary conditions) are studied numerically. The Carreau-Yasuda model is used to provide further insight into the problem of the slip behavior of non-Newtonian fluids having a decreasing viscosity with a shear rate increase. This feature is demonstrated to cause a nonlinear velocity profile leading to the apparent slip. The corresponding transverse and longitudinal apparent slip lengths of a striped texture are found to be noticeably larger than the respective effective slip lengths of Newtonian liquids in microchannels of various thicknesses and surface fractions of the slip domains. The viscosity distribution of the shear thinning fluid over the superhydrophobic wall is carefully investigated to describe the mechanism of the apparent slip. Nonmonotonic behavior of the apparent slip length as a function of the applied shear rate is revealed. This important property of shear thinning fluids is considered to be sensitive to the steepness of the viscosity flow curve, thus providing a way to decrease considerably the flow resistance in microchannels.

Forces and torques on rigid inclusions in an elastic environment: Resulting matrix-mediated interactions, displacements, and rotations.

Embedding rigid inclusions into elastic matrix materials is a procedure of high practical relevance, for instance, for the fabrication of elastic composite materials. We theoretically analyze the following situation. Rigid spherical inclusions are enclosed by a homogeneous elastic medium under stick boundary conditions. Forces and torques are directly imposed from outside onto the inclusions or are externally induced between them. The inclusions respond to these forces and torques by translations and rotations against the surrounding elastic matrix. This leads to elastic matrix deformations, and in turn results in mutual long-ranged matrix-mediated interactions between the inclusions. Adapting a well-known approach from low-Reynolds-number hydrodynamics, we explicitly calculate the displacements and rotations of the inclusions from the externally imposed or induced forces and torques. Analytical expressions are presented as a function of the inclusion configuration in terms of displaceability and rotateability matrices. The role of the elastic environment is implicitly included in these relations. That is, the resulting expressions allow a calculation of the induced displacements and rotations directly from the inclusion configuration, without having to explicitly determine the deformations of the elastic environment. In contrast to the hydrodynamic case, compressibility of the surrounding medium is readily taken into account. We present the complete derivation based on the underlying equations of linear elasticity theory. In the future, the method will, for example, be helpful to characterize the behavior of externally tunable elastic composite materials, to accelerate numerical approaches, as well as to improve the quantitative interpretation of microrheological results.