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Capillary sorting of fiber suspensions by dip coating
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Abstract
Sorting elongated anisotropic particles, such as fibers, dispersed in suspensions poses significant challenges as they present two characteristic dimensions: length and diameter. Fibers in suspension usually align with the flow, leading to diameter-based filtration when passing through a sieve. Modifying the flow conditions by introducing more mixing so that fibers are arbitrarily oriented can lead to sorting by diameter and length simultaneously, resulting in a lower filtration quality. In this paper, we demonstrate that capillary filtration by dip coating can be utilized to selectively sort fibers by length or by diameter in a controlled manner. Using the withdrawal of a flat substrate from a fiber suspension, we demonstrate that fibers are primarily sorted by their diameters. When considering cylindrical substrates, fibers can be sorted by length under appropriate conditions due to the orientation adopted by the fibers during their entrainment. We report guidelines for designing this filtration process and obtaining good sorting efficiency.
... fiber suspensions. 51,52 Understanding the full role of surfactants in particle entrainment in a shear-thinning medium is also an interesting extension, since the concentration near the CMC have been shown to exihibit transitions in the flow structure of Newtonian solutions. 53 ...
Dip coating a planar substrate with a suspension of particles in a shear-thinning liquid will entrain particles in the liquid film, facilitating filtration and sorting of particles.
Micro-scale inorganic particles (d > 1 µm) have reduced surface area and higher density, making them negatively buoyant in most dip-coating mixtures. Their controlled delivery in hard-to-reach places through entrainment is possible but challenging due to the density mismatch between them and the liquid matrix called liquid carrier system (LCS). In this work, the particle transfer mechanism from the complex density mismatching mixture was investigated. The LCS solution was prepared and optimized using a polymer binder and an evaporating solvent. The inorganic particles were dispersed in the LCS by stirring at the just suspending speed to maintain the pseudo suspension characteristics for the heterogeneous mixture. The effect of solid loading and the binder volume fraction on solid transfer has been reported at room temperature. Two coating regimes are observed (i) heterogeneous coating where particle clusters are formed at a low capillary number and (ii) effective viscous regime, where full coverage can be observed on the substrate. ‘Zero’ particle entrainment was not observed even at a low capillary number of the mixture, which can be attributed to the presence of the binder and hydrodynamic flow of the particles due to the stirring of the mixture. The critical film thickness for particle entrainment is h∗=0.16a for 6.5% binder and h∗=0.26a for 10.5% binder, which are smaller than previously reported in literature. Furthermore, the transferred particle matrices closely follow the analytical expression (modified LLD) of density matching suspension which demonstrate that the density mismatch effect can be neutralized with the stirring energy. The findings of this research will help to understand this high-volume solid transfer technique and develop novel manufacturing processes.
Significance
The pinch-off of a liquid drop extruded from a nozzle is a canonical situation that involves a series of self-similar regimes ending in a finite-time singularity. This configuration allows for exploring capillary flows over a large range of scales. In the case of suspension drops, the presence of particles breaks the self-similarity by introducing a length scale that can be much larger than the particle diameter. This length scale is a signature of the heterogeneities and delimitates a regime, in which a continuum approach of a suspension can be used from a regime where the discrete nature of the particles is involved.
Dip coating consists of withdrawing a substrate from a bath to coat it with a thin liquid layer. This process is well understood for homogeneous fluids, but heterogeneities, such as particles dispersed in liquid, lead to more complex situations. Indeed, particles introduce a new length scale, their size, in addition to the thickness of the coating film. Recent studies have shown that, at first order, the thickness of the coating film for monodisperse particles can be captured by an effective capillary number based on the viscosity of the suspension, providing that the film is thicker than the particle diameter. However, suspensions involved in most practical applications are polydisperse, characterized by a wide range of particle sizes, introducing additional length scales. In this study, we investigate the dip coating of suspensions having a bimodal size distribution of particles. We show that the effective viscosity approach is still valid in the regime where the coating film is thicker than the diameter of the largest particles, although bidisperse suspensions are less viscous than monodisperse suspensions of the same solid fraction. We also characterize the intermediate regime that consists of a heterogeneous coating layer and where the composition of the film is different from the composition of the bath. A model to predict the probability of entraining the particles in the liquid film depending on their sizes is proposed and captures our measurements. In this regime, corresponding to a specific range of withdrawal velocities, capillarity filters the large particles out of the film.
When a droplet is generated, the ligament connecting the drop to the nozzle thins down and eventually pinches off. Adding solid particles to the liquid phase leads to a more complex dynamic, notably by increasing the shear viscosity. Moreover, it introduces an additional length scale to the system, the diameter of the particles, which eventually becomes comparable to the diameter of the ligament. In this paper, we experimentally investigate the thinning and pinch-off of drops of suspensions with two different sizes of particles. We characterize the thinning for different particle size ratios and different proportions of small particles. Long before the pinch-off, the thinning rate is that of an equivalent liquid whose viscosity is that of the suspension. Later, when the ligament thickness approaches the size of the large particles, the thinning accelerates and leads to an early pinch-off. We explain how the bidisperse particle size distribution lowers the viscosity by making the packing more efficient, which speeds up the thinning. This result can be used to predict the dynamics of droplet formation with bidisperse suspensions.
The withdrawal of a liquid or the translation of a liquid slug in a capillary tube leads to the deposition of a thin film on the inner wall. When particles or contaminants are present in the liquid, they deposit and contaminate the tube if the liquid film is sufficiently thick. In this article, we experimentally investigate the condition under which particles are deposited during the air invasion in a capillary tube initially filled with a dilute suspension. We show that the entrainment of particles in the film is controlled by the ratio of the particle and the tube radii and the capillary number associated with the front velocity. We also develop a model which suggests optimal operating conditions to avoid contamination during the withdrawal of a suspension from a tube. This deposition mechanism can also be leveraged in coating processes by controlling the deposition of particles on the inner walls of channels.
A fibre withdrawn from a bath of a dilute particulate suspension exhibits different coating regimes depending on the physical properties of the fluid, the withdrawal speed, the particle sizes and the radius of the fibre. Our experiments indicate that only the liquid without particles is entrained for thin coating films. Beyond a threshold capillary number, the fibre is coated by a liquid film with entrained particles. We systematically characterize the role of the capillary number, the particle size and the fibre radius on the threshold speed for particle entrainment. We discuss the boundary between these two regimes and show that the thickness of the liquid film at the stagnation point controls the entrainment process. The radius of the fibre provides a new degree of control in capillary filtering, allowing greater control over the size of the particles entrained in the film.
The fragmentation of liquid sheets produces a collection of droplets. The size distribution of the droplets has a considerable impact on the coating efficiency of sprays and the transport of contaminants. Although many processes commonly used particulate suspensions, the influence of the particles on the spreading dynamics of the sheet and its subsequent fragmentation has so far been considered negligible. In this paper, we consider experimentally a transient suspension sheet that expands radially. We characterize the influence of the particles on the dynamics of the liquid sheet and the fragmentation process. We highlight that the presence of particles modifies the thickness and reduces the stability of the liquid sheet. Our study suggests that particles can significantly modify the dynamics of liquid films through capillary effects, even for volume fractions much smaller than the maximum packing.
In this letter, we describe the capillary sorting of particles by size based on dip coating. A substrate withdrawn from a liquid bath entrains a coating whose thickness depends on the withdrawal speed and the liquid properties. If the coating material contains particles, they will only be entrained when the viscous force pulling them with the substrate overcomes the opposing capillary force at the deformable meniscus. This force threshold occurs at different liquid thicknesses for particles of different sizes. Here, we show that this difference can be used to separate small particles from a mixed suspension through capillary filtration. In a bidisperse suspension, we observe three distinct filtration regimes. At low capillary numbers, Ca, no particles are entrained in the liquid coating. At high Ca, all particle sizes are entrained. For a range of capillary numbers between these two extremes, only the smallest particles are entrained while the larger ones remain in the reservoir. We explain how this technique can be applied to polydisperse suspension. We also provide an estimate of the range of capillary number to separate particles of given sizes. The combination of this technique with the scalability and robustness of dip coating makes it a promising candidate for high-throughput separation or purification of industrial and biomedical suspensions.
An object withdrawn from a liquid bath is coated with a thin layer of liquid. Along with the liquid, impurities such as particles present in the bath can be transferred to the withdrawn substrate. Entrained particles locally modify the thickness of the film, hence altering the quality and properties of the coating. In this study, we show that it is possible to entrain the liquid alone and avoid contamination of the substrate, at sufficiently low withdrawal velocity in diluted suspensions. Using a model system consisting of a plate exiting a liquid bath, we observe that particles can remain trapped in the meniscus which exerts a resistive capillary force to the entrainment. We characterize different entrainment regimes as the withdrawal velocity increases: from a pure liquid film, to a liquid film containing clusters of particles, and eventually individual particles. This capillary filtration is an effective barrier against the contamination of substrates withdrawn from a polluted bath and finds application against biocontamination.
The coating of a plate withdrawn from a bath of a suspension of non-Brownian, monodisperse and neutrally buoyant spherical particles suspended in a Newtonian liquid has been studied. Using laser profilometry, particle tracking and local sample weighing we have quantified the thickness h and the particle content of the film for various particle diameters d and volume fractions (0.10 φ 0.50). Three coating regimes have been observed as the withdrawal velocity is increased: (i) no particle entrainment (h d), (ii) a monolayer of particles (h ∼ d), and (iii) a thick film (h d), where the suspension behaves as an effective viscous fluid following the Landau-Levich-Derjaguin law. We discuss the boundaries between these regimes, as well as the evolution of the liquid and solid content of the coating over the whole range of withdrawal capillary number and volume fractions.
The effects of particle shape and operating conditions on the sieving process were experimentally studied and the passing characteristics were discussed. Cylinder particles of unity in aspect ratio were batchwise sieved by a reciprocal shifter. The retention on a woven wire sieve of square mesh was measured and the sieving rates were obtained. The behaviour of retention with sieving time were classified into two types according as the cylinder diameter D was near the mesh size a or D « a. In the latter case, the cylinder was easily screenable and the retention exponentially decreased with sieving time. The sieving rate coefficient was independent of the initial charge. On the other hand, in the former case, the sieving rate was initially influenced by the particle mass charged on the sieve and the sieving process deviated from the exponential decrease. The sieving rate was quantitatively associated with the geometrical shape of cylinders. The sieving rate coefficients of easily screenable cylinders or in the earlier period of sieving were inversely proportional to the third power of the cylinder length. The probabilities of the cylinder passing through an opening were calculated by use of geometrical probabilities and compared with the experimental results. The number of trials of passing per unit time was almost inversely proportional to the cylinder length. Furthermore, the sieving characteristics of a set of sieves were analyzed on the basis of the rate coefficient obtained previously. It was possible to predict the change of each retention with sieving time, sufficiently. The correlation between the equivalent spherical diameter of cylinder and 50%-separation diameter measured by the multistage sieving gave that the separation diameter mainly depended on the cylinder diameter. In the vicinity of centrifugal effect of 1.3, the sieving rate was maximum and the empirical relationship presented by previous workers for the optimum shifting conditions was applicable to the cylinder particles.
† This report was originally printed in J. Soc. Powder Technology, Japan, 20(1), 8-14 (1983) in Japanese, before being translated into English by KONA Editorial Committee with the permission of the editorial committee of the Soc. Powder Technology, Japan.
We report on a microfluidic method that allows measurement of a small concentration of large contaminants in suspensions of solid micrometer-scale particles. To perform the measurement, we flow the colloidal suspension through a series of constrictions, i.e., a microchannel of varying cross-section. We show and quantify the role of large contaminants in the formation of clogs at a constriction and the growth of the resulting filter cake. By measuring the time interval between two clogging events in an array of parallel microchannels, we are able to estimate the concentration of contaminants whose size is selected by the geometry of the microfluidic device. This technique for characterizing colloidal suspensions offers a versatile and rapid tool to explore the role of contaminants on the properties of the suspensions.
The formation of drops of particulate suspensions composed of spherical, neutrally buoyant,
noncolloidal particles in a viscous liquid is examined experimentally. The suspensions are
investigated over a range of particle volume fractions, #!n$dp
3/6 where n is the particle number
density and dp is the particle diameter (dp!212– 250 %m in most of the experiments", and for flow
through three tubes of outer diameters d!0.16, 0.32 and 0.64 cm; the corresponding inner
diameters are 0.10, 0.22, and 0.53 cm. Drop formation in the dripping mode and the transition from
dripping to jetting are investigated. In the dripping mode, the behavior of low-# suspensions (#
&0.10) is markedly different from that of higher-# suspensions (0.15&#&0.40), with a transition
in the qualitative behavior over a relatively narrow range of concentrations. Pinch-off structures for
low-# suspensions are similar to that of the pure liquid, consisting of a long slender liquid thread
connecting a hemispherical cap of liquid at the orifice to the nearly spherical forming drop; the
structures differ from those of pure liquids in that individual particles, or groups of particles, can be
captured in the thread at rupture, leading to new modes of satellite drop formation. At higher #, the
presence of large numbers of particles in the thinning thread during necking results in thick conelike
structures termed ‘‘spindles’’ owing to this labeling of structures of similar geometry observed in
electrohydrodynamic sprays. Particles are found to substantially suppress the number of satellite
drops at higher #, but the few satellite drops produced are much larger than observed in pure liquid
drop formation. The transition from dripping to jetting occurs at a smaller flow rate at finite #, with
much shorter coherent lengths after jetting than in the case of a pure liquid. The transition becomes
less abrupt and more difficult to identify at the highest concentrations examined.
Hypothesis:
The dip coating of suspensions made of monodisperse non-Brownian spherical particles dispersed in a Newtonian fluid leads to different coating regimes depending on the ratio of the particle diameter to the thickness of the film entrained on the substrate. In particular, dilute particles dispersed in the liquid are entrained only above a threshold value of film thickness. In the case of anisotropic particles, in particular fibers, the smallest characteristic dimension will control the entrainment of the particle. Furthermore, it is possible to control the orientation of the anisotropic particles depending on the substrate geometry. In the thick film regime, the Landau-Levich-Derjaguin model remains valid if one account for the change in viscosity.
Experiment:
To test the hypotheses, we performed dip-coating experiments with dilute suspensions of non-Brownian fibers with different length-to-diameter aspect ratios. We characterize the number of fibers entrained on the surface of the substrate as a function of the withdrawal velocity, allowing us to estimate a threshold capillary number below which all the particles remain in the liquid bath. Besides, we measure the angular distribution of the entrained fibers for two different substrate geometries: flat plates and cylindrical rods. We then measure the film thickness for more concentrated fiber suspensions.
Findings:
The entrainment of the fibers on a flat plate and a cylindrical rod is primarily controlled by the smaller characteristic length of the fibers: their diameter. At first order, the entrainment threshold scales similarly to that of spherical particles. The length of the fibers only appears to have a minor influence on the entrainment threshold. No preferential alignment is observed for non-Brownian fibers on a flat plate, except for very thin films, whereas the fibers tend to align themselves along the axis of a cylindrical rod for a large enough ratio of the fiber length to the radius of the cylindrical rod. The Landau-Levich-Derjaguin law is recovered for more concentrated suspension by introducing an effective capillary number accounting for the change in viscosity.
The displacement of a suspension of particles by an immiscible fluid in a capillary tube or in porous media is a canonical configuration that finds application in a large number of natural and industrial applications, including water purification, dispersion of colloids and microplastics, coating and functionalization of tubings. The influence of particles dispersed in the fluid on the interfacial dynamics and on the properties of the liquid film left behind remain poorly understood. Here, we study the deposition of a coating film on the walls of a capillary tube induced by the translation of a suspension plug pushed by air. We identify the different deposition regimes as a function of the translation speed of the plug, the particle size, and the volume fraction of the suspension. The thickness of the coating film is characterized, and we show that similarly to dip coating, three coating regimes are observed, liquid only, heterogeneous, and thick films. We also show that, at first order, the thickness of films thicker than the particle diameter can be predicted using the effective viscosity of the suspension. Nevertheless, we also report that for large particles and concentrated suspensions, a shear-induced migration mechanism leads to local variations in volume fraction and modifies the deposited film thickness and composition.
A thin viscous layer is found over a substrate when it is immersed in a polymer solution. The layer thickness depends on the polymer and solvent type, their volume fraction, and the substrate. If the liquid solution contains particles, they are entrapped on the viscous polymer layer, acting as the binder. The trade-off between the viscous force and the centrifugal force on the particle determines the entrapment. Furthermore, the size of entrained particles are dictated by the binder concentration of the solution., A particle filtration technique is presented using the entrapment phenomenon from a poly-disperse mixture. A dimensionless number called the entrapment factor is introduced to correlate the particle entrapment with various parameters. By changing the entrapment factor, three distinct entrapment regimes are achieved and explained from a poly-disperse mixture. The experimental result shows that entrapped particles become larger as the factor increases, which can be controlled with multiple parameters of the dipping process. The proposed technique can lead to a filtration process of the wide-range poly-disperse particle mixture over the capillary filtration processes.
The formation of aggregates in solutions of monoclonal antibodies is difficult to prevent. Even if the occurrence of large aggregates is rare, their existence can lead to partial or total clogging of constrictions in injection devices, with drastic effects on drug delivery. Little is known on the origin and characteristics of such clogging events. Here we investigate a microfluidic model system to gain fundamental understanding of the clogging of constrictions by monoclonal antibody aggregates. Highly concentrated solutions of monoclonal antibodies were used to create protein aggregates (larger than m) using mechanical or heat stress. We show that clogging occurs when aggregates reach the size of the constriction and that clogs can in some cases be released by increasing the applied pressure. This indicates the important role of protein aggregate deformability. We perform systematic experiments for different relative aggregate sizes and applied pressures, and measure the resulting flow-rate. This allows us to present first in-situ estimates of an effective Young's modulus. Despite their different shapes and densities, we can predict the number of clogging events for a given constriction size from the aggregate size distribution measured by Flow Imaging Microscopy (MFI). In addition our device can detect the occurrence of very rare big aggregates often overlooked by other detection methods.
Shear reversal is a smart rheological protocol allowing inspection of the contribution of the nonhydrodynamic contact forces between particles on the rheology of concentrated suspensions. In this paper, this protocol is applied to concentrated suspensions of rigid fibers dispersed in a Newtonian liquid at three aspect ratios ranging from 10 to 31 and various concentrations. When the shear direction is reversed, a low viscosity pseudoplateau is observed and we show that the value of the viscosity at pseudoplateau is close to the steady-state viscosity value expected for semidilute suspensions where the fibers interact mainly via hydrodynamic forces, while the contact interactions are released for a short time just after shear reversal. The low viscosity pseudoplateau is followed by a viscosity overshoot that is interpreted in terms of the interplay between contact interparticle interactions (that reappear at the end of the low viscosity plateau) and fiber reorientation. In particular, we show that the accumulated strain at which the overshoot occurs is controlled by a strain scale associated with the reconstruction of the contact network. After the viscosity overshoot, the viscosity decreases and reaches the steady value that it had before shear reversal.
Withdrawing a plate from a suspension leads to the entrainment of a coating layer of fluid and particles on the solid surface. In this article, we study the Landau-Levich problem in the case of a suspension of non-Brownian particles at moderate volume fraction 10% < ϕ < 41%. We observe different regimes depending on the withdrawal velocity U, the volume fraction of the suspension ϕ, and the diameter of the particles 2a. Our results exhibit three coating regimes. (i) At small enough capillary number Ca, no particles are entrained, and only a liquid film coats the plate. (ii) At large capillary number, we observe that the thickness of the entrained film of suspension is captured by the Landau-Levich law using the effective viscosity of the suspension η(ϕ). (iii) At intermediate capillary numbers, the situation becomes more complicated with a heterogeneous coating on the substrate. We rationalize our experimental findings by providing the domain of existence of these three regimes as a function of the fluid and particles properties.
The pinch-off of a capillary thread is studied at large Ohnesorge number for non-Brownian, neutrally buoyant, mono-disperse, rigid, spherical particles suspended in a Newtonian liquid with viscosity \unicode[STIX]{x1D702}_{0} and surface tension \unicode[STIX]{x1D70E} . Reproducible pinch-off dynamics is obtained by letting a drop coalesce with a bath. The bridge shape and time evolution of the neck diameter, , are studied for varied particle size d , volume fraction \unicode[STIX]{x1D719} and liquid contact angle \unicode[STIX]{x1D703} . Two successive regimes are identified: (i) a first effective-viscous-fluid regime which only depends upon \unicode[STIX]{x1D719} and (ii) a subsequent discrete regime, depending both on d and \unicode[STIX]{x1D719} , in which the thinning localises at the neck and accelerates continuously. In the first regime, the suspension behaves as an effective viscous fluid and the dynamics is solely characterised by the effective viscosity of the suspension, \unicode[STIX]{x1D702}_{e}\sim -\unicode[STIX]{x1D70E}/{\dot{h}}_{\mathit{min}} , which agrees closely with the steady shear viscosity measured in a conventional rheometer and diverges as (\unicode[STIX]{x1D719}_{c}-\unicode[STIX]{x1D719})^{-2} at the same critical particle volume fraction, \unicode[STIX]{x1D719}_{c} . For \unicode[STIX]{x1D719}\gtrsim 35\,\% , the thinning rate is found to increase by a factor of order one when the flow becomes purely extensional, suggesting non-Newtonian effects. The discrete regime is observed from a transition neck diameter, h_{\mathit{min}}\equiv h^{\ast }\sim d\,(\unicode[STIX]{x1D719}_{c}-\unicode[STIX]{x1D719})^{-1/3} , down to , where the thinning rate recovers the value obtained for the pure interstitial fluid, \unicode[STIX]{x1D70E}/\unicode[STIX]{x1D702}_{0} , and lasts t^{\ast }\sim \unicode[STIX]{x1D702}_{e}h^{\ast }/\unicode[STIX]{x1D70E} .
The dynamics and rheology of suspensions of rigid, non-Brownian fibers in Newtonian fluids are reviewed. Experiments, theories, and computer simulations are considered, with an emphasis on suspensions at semidilute and concentrated conditions. In these suspensions, interactions between the particles strongly influence the microstructure and rheological properties of the suspension. The interactions can arise from hydrodynamic disturbances, giving multibody interactions at long ranges and pairwise lubrication forces over short distances. For concentrated suspensions, additional interactions due to excluded volume (contacts) and adhesive forces are addressed. The relative importance of the various interactions as a function of fiber concentration is assessed.
A solid withdrawn from a liquid bath entrains a film. In this review, after recalling the predictions and results for pure Newtonian liquids coated on simple solids, we analyze the deviations to this ideal case exploring successively three potential sources of complexity: the liquid-air interface, the bulk rheological properties of the liquid and the mechanical or chemical properties of the solid. For these different complexities, we show that significant effects on the film thickness are observed experimentally and we summarize the theoretical analysis presented in the literature, which attempt to rationalize these measurements.
The transport of suspensions of microparticles in confined environments is associated with complex phenomena at the interface of fluid mechanics and soft matter. Indeed, the deposition and assembly of particles under flow involve hydrodynamic, steric and colloidal forces, and can lead to the clogging of microchannels. The formation of clogs dramatically alters the performance of both natural and engineered systems, effectively limiting the use of microfluidic technology. While the fouling of porous filters has been studied at the macroscopic level, it is only recently that the formation of clogs has been considered at the pore-scale, using microfluidic devices. In this review, we present the clogging mechanisms recently reported for suspension flows of colloidal particles and for biofluids in microfluidic channels, including sieving, bridging and aggregation of particles. We discuss the technological implications of the clogging of microchannels and the schemes that leverage the formation of clogs. We finally consider some of the outstanding challenges involving clogging in human health, which could be tackled with microfluidic methods.
An extensive body of experimental work has proven the validity of the analysis of Landau and Levich, who were the first to determine theoretically the thickness of the film deposited by the withdrawal of a flat substrate from a bath of liquid with a clean interface. However, there are a number of experimental investigations that have shown that surfactants in the liquid may result in a thickening of the deposited film. Marangoni phenomena have usually been considered responsible for this effect. However, some careful experiments and numerical simulations reported in the literature seemed to rule out this view as the cause of the observed behavior. Despite all these studies and the number of reports of film thickening, an experimental study of the flow field close to the coated substrate in the presence of surfactants has never been undertaken. In this paper we will present a set of flow visualization experiments on coating of a planar substrate in the range of capillary numbers 10−4 ≲ Ca ≲ 10−3 for sodium dodecyl sulfate solutions with bulk concentrations of 0.25 CMC ⩽ C ⩽ 5.0 CMC (critical micelle concentration). It was evident during experiments that the flow field near the meniscus region exhibits patterns that can only be explained with a stagnation point residing in the bulk and not at the interface. As opposed to patterns with an interfacial stagnation point, the observed flow fields allow for the increase in film thickness due to the presence of surfactants compared to the clean interface case.
VARIOUS measures of the size of irregularly shaped particles as seen in profile under the microscope have been used, chosen according to their theoretical significance or practical ease of measurement. These include, using Heywood's notation1,2: (i) the diameter of the circle of equal area, d; (ii) the diameter of the circle of equal perimeter, D; (iii) the length of line bisecting the profile area (Martin's statistical diameter3), M; and (iv) the perpendicular distance between parallel tangents touching opposite sides of the profile (Feret's statistical diameter4), F. M and F are determined for randomly oriented particles, thus giving an average value over all possible orientations. d is usually regarded as the ideal measure of particles seen in profile, but is somewhat difficult to determine experimentally with precision. It is, however, common practice when sizing very small particles to estimate d visually by comparing them with standard reference circles on a Patterson and Cawood or similar type of eyepiece graticule5,6. M and F are convenient to measure in practice with aid of an eyepiece scale or filar micrometer, and have been extensively used by various Workers. D, or rather the ratio D/d, termed by Heywood1 the `contour ratio', and its reciprocal called the `degree of circularity' by Wadell7, have been used in discussing the shape and hydrodynamical properties of particles. For these purposes D has usually been determined by direct perimeter measurement of the projected images of particles. It does not appear to have been adopted intentionally in any work known to me as a direct single measure of particle size.
The problem of determining the thickness of the dragged layer as a function of the speed of the motion of the film and of parameters characteristic of the properties of the fluid is of essential interest for practice. In the chapter, the thickness of the layer and the quantity of fluid carried along when pulling an infinite plate out of a vessel, which is sufficiently large to permit the neglecting of the effect of its walls and of the edges of the plate, is evaluated. The case of low velocity of motion of the plate is considered. In this case, all the surface of the liquid may be separated into two independent regions: (1) the region of the surface situated high above the meniscus and directly dragged by the plate, where the surface of liquid may be taken to be nearly parallel to the plate surface and (2) the region of the meniscus of liquid. The solutions of hydrodynamical equations in both independent regions are presented in the chapter and then both of the solutions that are found are connected.
We study the hydrodynamics of dip coating from a suspension and report a mechanism for colloidal assembly and pattern formation on smooth substrates. Below a critical withdrawal speed where the coating film is thinner than the particle diameter, capillary forces induced by deformation of the free surface prevent the convective transport of single particles through the meniscus beneath the film. Capillary-induced forces are balanced by hydrodynamic drag only after a minimum number of particles assemble within the meniscus. The particle assembly can thus enter the thin film where it moves at nearly the withdrawal speed and rapidly separates from the next assembly. The interplay between hydrodynamic and capillary forces produces periodic and regular structures below a critical ratio Ca^{2/3}/sqrt[Bo]<0.7, where Ca and Bo are the capillary and Bond numbers, respectively. An analytical model and numerical simulations are presented for the case of two-dimensional flow with circular particles in suspension. The hydrodynamically driven assembly documented here is consistent with stripe pattern formations observed experimentally in dip coating.
▪ Abstract We discuss the thickness of the liquid layer entrained by a solid drawn out of a bath, focusing on the case where the solid is a fiber or a wire. Slow withdrawals out of a pure or a complex fluid are described as well as quick coatings. We specify the general laws of entrainment and stress the cases where the fiber curvature plays a role. We finally give an overview on the further evolution of the coated film.
The Laplace equation for the pressure drop across curved liquid-gas interfaces is applied to the solution of the profile of a static liquid meniscus on the outside of a wire of circular cross-section. The resulting differential equation is integrated numerically, an operation complicated by the existence of boundary conditions at two points making a trial-and-error solution necessary. The accuracy of the solution is substantiated by comparison of computed profiles with experiments in which menisci of a blue dye in water are photographed clinging to the outside of brass wires, whose diameters lie within the range of technological interest.
A theory of the amount of liquid entrained by cylinders upon withdrawal from liquid baths is derived for a wide range of cylinder radii. The theory is based on matching curvatures for static and dynamic menisci. Predicted values are expressed as the effect of the dimensionless wire radius (Goucher number) and dimensionless withdrawal speed (capillary number) on the dimensionless flux.
The theory was verified experimentally for all wire radii by removing short cylinders from oily fluids and with other information. The fluids used included kerosene, mineral oil, motor oil, and glycerine, with viscosities from 2 to 500 centipoise; Goucher numbers ranged from 0.05 to 1.2. Deviations, which were noted at high capillary numbers where velocity gradients become appreciable, indicated that the theory is a plug flow or low speed theory. Also discussed are differences found with water and the conditions under which films coalesced into droplets.
We experimentally study the detachment of drops of granular suspensions using
a density matched model suspension with varying volume fraction ({\phi} = 15%
to 55%) and grain diameter (d = 20 {\mu}m to 140 {\mu}m). We show that at the
beginning of the detachment process, the suspensions behave as an effective
fluid. The detachment dynamics in this regime can be entirely described by the
shear viscosity of the suspension. At later stages of the detachment the
dynamics become independent of the volume fraction and are found to be
identical to the dynamics of the interstitial fluid. Surprisingly, visual
observation reveals that at this stage particles are still present in the neck.
We suspect rearrangements of particles to locally free the neck of grains,
causing the observed dynamics. Close to the final pinch off, the detachment of
the suspensions is further accelerated, compared to the dynamics of pure
interstitial fluid. This acceleration might be due to the fact that the neck
diameter gets of the order of magnitude of the size of the grains and a
continuous thinning of the liquid thread is not possible any more. The
crossover between the different detachment regimes is function of the grain
size and the initial volume fraction. We characterize the overall acceleration
as a function of the grain size and volume fraction.
In this work we present an experimental study of deviations from the classical Landau-Levich law in the problem of dip coating. Among the examined causes leading to deviations are the nature of the liquid-gas and liquid-solid interfaces. The thickness of the coating film created by withdrawal of a plate from a bath was measured gravimetrically over a wide range of capillary numbers for both smooth and well-characterized rough substrates, and for clean and surfactant interface cases. In view of the dependence of the lifetime of a film on the type of liquid and substrate, and liquid-gas and liquid-solid interfaces, we characterized the range of measurability of the film thickness in the parameter space defined by the withdrawal capillary number, the surfactant concentration, and substrate roughness size. We then study experimentally the effect of a film thickening due to the presence of surfactants. Our recent theory based on a purely hydrodynamic role of the surface active substance suggests that there is a sorption-controlled coating regime in which Marangoni effects should lead to film thinning. However, our experiments conducted in this regime demonstrate film thickening, calling into question the conventional wisdom, which is that Marangoni stresses (as accounted by the conventional interfacial boundary conditions) lead to film thickening. Next we examine the effect of well-characterized substrate roughness on the coated film thickness, which also reveals its influence on wetting-related processes and an effective boundary condition at the wall. In particular, it is found that roughness results in a significant thickening of the film relative to that on a smooth substrate and a different power of capillary number than the classical Landau-Levich law.
Sedimentation field-flow fractionation (SdFFF) with UV detection is used to systematically investigate the effect of traditional membrane filtration and centrifugation procedures on the isolation of specific size fractions from soil suspensions. Both procedures were used to isolate the nominal <0.45 and <0.2 microm fractions from a clay soil suspension. Results showed that the membrane filtration approach seriously underestimated the total mass of particulate matter present as compared to the centrifugation approach. This has serious implications forthe interpretation of results for "colloidal" and "soluble" fractions from soil suspensions and other environmental matrices obtained using the standard membrane approach. The results also show that sedimentation FFF has great potential as a robust and relatively mild technology for studying size distributions in the "colloidal" range for soil suspensions and other aquatic matrices.
- M Haw
M. Haw, Physical review letters 92, 185506 (2004).
- E Dressaire
- A Sauret
E. Dressaire and A. Sauret, Soft matter 13, 37 (2017).
- C Duchêne
- V Filipe
- S Huille
- A Lindner
C. Duchêne, V. Filipe, S. Huille, and A. Lindner, Soft matter
16, 921 (2020).
- B M Dincau
- E Mai
- Q Magdelaine
- J Lee
- M Bazant
- A Sauret
B. M. Dincau, E. Mai, Q. Magdelaine, J. Lee, M. Bazant, and
A. Sauret, Journal of fluid mechanics 903, A38 (2020).
- I Khalil
- B Khoda
I. Khalil and B. Khoda, Journal of Manufacturing Science and
Engineering 144, 054502 (2022).
- E Rio
- F Boulogne
E. Rio and F. Boulogne, Advances in Colloid and Interface Science 247, 100 (2017).
- L Landau
- B Levich
L. Landau and B. Levich, Acta Physicochimica URSS 17, 42
(1942).