Washburn Equation Research Articles

Absorption of millimeter- and micrometer-sized droplets on nylon powder

Absorption of droplet by the powder media is a ubiquitous phenomenon in many natural and engineering processes, in which the diameter of droplets ranges from tens of micrometers (μm) to a few millimeters (mm) depending on the specific condition. In this study, we investigate the absorption process following the impact of the droplet with radii varying from mm- to μm-scale on nylon powder substrate. The fluid properties are varied by mixing the deionized water with isopropyl alcohol at different volume ratios. A syringe pipette tip is used to generate droplets of diameter ranging from 2.08 to 2.30 mm, whereas a drop-on-demand inkjet device is used to dispense droplets of diameter ranging from 88 to 140 μm. The absorption process of the single droplet is captured by a high-speed imaging system. We find that for mm- and μm-droplets, there exists different critical surface tension below which the imbibition into the nylon powder can occur. With the proposed nondimensionalization process, the scattered data of dimensionless absorbed volume V and dimensionless time τ for different fluids generally collapse into a power law relation V ~ τα, where the exponent α for mm-sized and μm-sized droplets is 0.856 and 0.518, respectively. The experimental data of μm droplets generally agree with the prediction of the Washburn equation and diffusion-type equation for the unidirectional capillary flow, i.e., V ~ τ0.5. However, the results of mm-droplets favor more the prediction of the 3D radial flow (i.e., a hemispherically advancing infiltration front inside the powder), V ~ τ.

Optimization for ultrafast capillary-driven flow in open rectangular microchannels

The Lucas−Washburn (LW) equation has been extensively used to predict the imbibition length of capillary-driven flow. Using the hydraulic diameter for non-circular cross section in the LW equation is widely accepted. Accordingly, the larger the hydraulic diameter, the higher the imbibition rate. However, the validity of the LW equation for the optimization of capillary-driven flow in open rectangular microchannels remains an open question. In this study, we propose a theoretical model to predict the imbibition rate of capillary-driven flow in open rectangular microchannels. To validate the model, we conducted a series of experiments on spontaneous capillary-driven flow of water in open rectangular Si microchannels. We fabricated Si microchannels with hydraulic diameters ranging from 18 to 375 µm to investigate the dependence of the imbibition rate on the hydraulic diameter. The proposed model was consistent with the experimental results. In the case of microchannels with a fixed height, a plot of the measured imbibition rate against the hydraulic diameter was well fitted by a downward-opening parabolic curve, indicating that the imbibition rate had an optimal value at the aspect ratio (width to height) of two. For a given channel width, the imbibition rate increased with the hydraulic diameter, and it leveled off at a large hydraulic diameter for aspect ratios below 0.5. An ultrafast imbibition time of 1.97 s over an entire length of 70 mm was achieved, corresponding to the optimal imbibition rates of 40 mm/s at imbibition length x1 = 34 mm and 19 mm/s at x2 = 68 mm among the experimental cases.

Imbibition dynamics and steady flows in graphene nanochannels with sparse geometric and chemical defects

Geometric and chemical defects are frequently found or created on smooth graphene for applications of nanofluidics. In this work, imbibition dynamics and steady flows of water in graphene nanochannels with sparse defects are explored by molecular dynamics. The water contact angle is raised slightly by geometric defects (hole and protrusion) but lowered significantly by chemical defects (hydroxyl and epoxide groups). In steady flows, the mean velocity and slip length are always reduced by sparse defects and the effect of chemical defects is more significant than that of geometric defects. Moreover, it is interesting to find that the velocity profile is plug-like for geometric defects but becomes parabolic for chemical defects, regardless of the slip length. Sparse defects on graphene nanoslits also affect the imbibition dynamics remarkably, which generally follows Washburn's equation with the slip length. For chemical defects, surface friction (slip length) dominates over the driving force associated with surface wettability (contact angle). Nonetheless, for protrusion defects, the stick-slip behavior caused by contact line pinning and thermal fluctuations can be observed. Our new and novel findings indicate that the defect nature is crucial in nanoscale flows and imbibition processes, which the conventional hydrodynamic theory fails to depict.

Influence of the Pore Radius on the Penetration Depth of Inks in Binder Jetting—A Modification of the Washburn Equation

In binder jetting (BJ), an ink is inserted layerwise into a powder bed to selectively bond the particles in the cross-section of a part. By predicting the penetration depth of the ink, the ideal layer thickness for BJ can be set. Each layer should be penetrated with ink. Insufficient penetration will result in a poor layer bond and a low strength of the part; over-penetration will impede a dimensionally accurate production, as the ink will leak from the sides of the part and unintentionally solidify the powder in these areas. The Washburn equation has been used for the calculation of the penetration depth in various fields, such as hydrology or with loose powders. However, a transfer to the BJ process is difficult due to the preferably compact powder bed and the fine particles. In more compact powder beds, the small radii with their greater capillary pressure and their distribution in the layer have a high influence on the penetration depth. This work shows an adaptation of the Washburn equation for powder beds in BJ and a new approach to determine the effective pore radius for calculating the penetration depth. A weighted pore radius was introduced, which accounts for the spatial distribution of the pores in the powder bed and the acting capillary pressure. The validation was performed with two different powders by experimentally simulating the BJ process through the infiltration of a drop into a powder bed. The weighted radius was used in the Washburn equation to calculate the penetration depth. The results were compared with those models from the literature and experimental data, and a good agreement between the calculation and the experiment was found.

Manganese silicide nanowires via metallic flux nanonucleation: growth mechanism and temperature-dependent resistivity

Mn5Si3 nanowires are believed to be the building blocks of the newest trends of flexible and stretchable devices in nanoelectronics. In this context , growing Mn5Si3 nanowires, as well as characterizing their electronic transport properties provide insight into their phenomenology. In this work, we report on the growth mechanism of Mn5Si3 nanowires produced by the metallic flux nanonucleation method, as well as the resistivity measurements of these nanostructures. Our calculation allows us, by using the Washburn equation for pore infiltration, to give a guess on why we obtain Mn-rich nanowires. In addition, some morphological aspects of the diameter-modulated Mn5Si3 nanowires were discussed based on the classical nucleation theory. From the resistivity measurements for the smallest diameter among the nanowires, we observed a significant reduction of around 37% of the phonons characteristic temperature by fitting the Bloch–Grünesein formula with other sources of scattering. Our results lead to a better understanding on the recent metallic flux nanonucleation growth method, as well as going a step further into the electronic transport properties of the Mn5Si3 nanowires.

Moisture Transport of Axial-Compression-Damaged Mortar and Concrete in Atmospheric Environment

The moisture transport of axial-compression-damaged mortar and concrete was experimentally and analytically studied in this paper. Five stress levels, i.e., 25%, 40%, 55%, 70%, and 85%, of the corresponding ultimate compressive strengths were selected for mortar and concrete specimens with the water cement ratio (w/c) of 0.5. Porosities and sorptivities of mortar or concrete before and after axial compression were measured and compared. Based on the Lucas–Washburn equation on absorption, the relationship between sorptivity and pore size distribution as well as porosity was established. A damage-representative radius was proposed to simply quantify the variation of pore characteristics of damaged mortar and concrete, and the moisture transport of axial-compression-damaged mortar and concrete could be predicted by summing the contributions to water absorption from the original pore system and the pore-equivalent microcrack system. It is shown that the porosities of mortar and concrete only slightly increase with the damage level, but the sorptivities are sensitive to axial compression damage, i.e., increasing nearly monotonically with the stress level from 0.3326 to 0.3533 mm/min0.5 for damaged mortar specimens (w/c = 0.5) and from 0.1970 to 0.2226 mm/min0.5 for damaged concrete specimens (w/c = 0.5). The increase trend became more apparent for both materials after a threshold of 40–55% of the corresponding ultimate compressive strengths, which is within the service load of structures, indicating that damage should be considered for chloride ions and water transport in concrete in the tidal zone. The predicted moisture diffusivities of damaged mortar and concrete show marginal difference from those of sound materials because the damage-representative radius could be underestimated due to elastic recovery of materials after unloading.

Implementation and Experiment of a Novel Molecular Spring Isolator

A molecular spring isolator comprising ZIF-8 with hydrophobic micropores, water, and a cylinder/piston unit is introduced. Liquid water can be intruded into the micropores of ZIF-8 under critical pressure. In the meantime, mechanical energy is stored. And once the external pressure is withdrawn, spontaneous capillary evaporation will occur. The stored energy, meanwhile, is released. This behavior is similar to a coil spring. The Laplace–Washburn equation is used to describe the force equilibrium of a single water column in a micropore. After that, the result of water infiltrating a great deal of hydrophobic pores is revealed by theoretical analysis and an experiment. An averaging method is devoted to computing the primary resonance response and its accuracy is proved by the fourth-order Runge–Kutta method. Furthermore, the energy transmissibility is used to estimate its vibration isolation capability. Finally, a vibration isolation test indicated that the inherent frequency of the isolator is as low as 1.3 Hz.

A new dynamic imbibition model for penny-shaped blind pores in shale gas well

Penny-shaped blind pores are common in shale, these pores with the original gas occurrence, including free gas and adsorbed gas, are seriously affected by a large amount of imbibed fracturing fluid. The competitive gas–water adsorption mechanism and the equations of gas-water phase pressures were analyzed statistically to detail the changes in gas–water distribution during imbibition. Based on the Lucas–Washburn equation, a new dynamic imbibition model is established for penny-shaped blind pores in shale gas reservoirs, which first considers the resistance of desorption of adsorbed gas and free gas and the factors of pore shape and fractal characteristics. The experimental validation, dimensionless-type curve, and common features of the new model regarding water imbibition are analyzed in detail. The results indicated that the proposed model is suitable for accurately predicting the imbibition in blind pores, and the dimensionless-type curve can simplify the calculation process for imbibition predictions. Factor analysis showed that penny-shaped pore width and length are two important factors for imbibition, the smaller width and the longer length, the more imbibition. Meanwhile, the increase of gas-water interfacial tension and displacement pressure is conducive to imbibition, the increase of initial gas pressure and desorption of adsorbed gas is obstructive to imbibition. The research is not only applicable in shale gas reservoir but also useful for geological hydrogen storage, carbon dioxide storage and geothermal development.

Wettability and Stability of Naproxen, Ibuprofen and/or Cyclosporine A/Silica Delivery Systems

The characteristics of the wetting process of the porous surface of silica gel when penetrated by base liquids (water and n-octane), ethanol and stable drug systems (naproxen, ibuprofen and cyclosporine A), as biologically active substances in two ethanol concentrations, were determined by the wetting rate vs. time. The tests were performed for contacted and non-contacted plates with the vapours of the wetting liquid. Thin-layer liquid chromatography was used to determine the penetration rate of the SiO2-coated plates, taking into account the linear dependence consistent with the Washburn equation. Additionally, the changes in the adhesive tension ΔG were determined for the tested drugs. Drug stability tests were conducted using the dynamic light scattering technique and microelectrophoresis. The penetration time of the plate depends on the properties and structure of the wetting liquid droplets. The types of interactions (dispersive, electrostatic and hydrogen bonding) formed between the silanol surface groups of the silica gel and the groups contained in the adsorbate particles are also very important factors. The greater the impact force, the slower the wetting process due to the strong penetration of the liquid into the pores of the substrate. The characteristics of the drug wetting/stability process may contribute to the development of their new forms, creating delivery systems with greater efficiency and lower side effects.

Spontaneous imbibition model for micro–nano–scale pores in shale gas reservoirs considering gas–water interaction

Spontaneous imbibition is the main factor that reduces the fracturing efficiency and damages gas production in shale gas wells, which is regulated by the gas–water interaction. Water (fracturing fluid) is imbibed spontaneously due to the characteristics of the water–rock interface, especially in shale with micro–nano–scale pores, while imbibition is not impeded but prevented by the pore structure resistance and original free gas compression as well as the increased fraction from the desorption of adsorbed gas. To accurately predict the fluid loss in a shale gas reservoir during fracturing, the imbibition pressures of the gas–water phase were comprehensively analyzed in this study, and the increased gas pressure from the desorption of adsorbed gas was first considered with other imbibition pressures, such as gas pressure from free gas compression, displacement pressure, capillary pressure, and osmotic pressure. By substituting the gas–water phase pressures into the Lucas–Washburn equation with the fractal characteristic of the capillary bundle, an imbibition model for micro–nano-scale pores in shale gas reservoirs considering gas–water interaction is established. The proposed model is compared with the traditional imbibition models and verified through imbibition experiments, and a case application to a shale gas well in the Sichuan Basin is undertaken. The results prove that the new fractal imbibition model performs better than the other models for imbibition estimation in reservoir shale, and it can accurately predict the fluid loss during the hydraulic fracturing process for shale gas wells.

An efficient solution to determine surface energy of powders and porous media: Application to untreated and treated lignin

The estimation of solid surface energy is generally obtained through the determination of liquid/solid contact angles. Many questions arise in the literature about the appropriate contact angle to use (dynamic, quasi-static or static, measured at a macro or a micro-scale). Theoretically, equilibrium contact angles should be inserted into equations to determine surface energy components. However, for powders or porous materials more issues arise, notably due to the imbibition of the liquid into the medium. An apparent contact angle can be obtained using wicking tests and the modified Washburn equation. Nonetheless, Washburn hypotheses are not always respected or no longer valid during wicking meaning that this procedure could not allow to determine contact angles. This study has the aim to propose a new simple method to determine the surface energy components of powders and porous media. An experimental protocol coupled to a modified Jurin law is proposed. This protocol was applied to lignin particles used as reinforcement in polylactic acid (PLA). Some treatments were performed on lignin to improve the adhesion with PLA and the modifications due to the treatments were characterized. Reliable surface energy and components of untreated and treated lignin were obtained revealing the efficiency of the proposed method.

Roles of energy dissipation and asymmetric wettability in spontaneous imbibition dynamics in a nanochannel

HypothesisThe imbibition dynamics is controlled by energy dissipation mechanisms and influenced by asymmetric wettability in a nanochannel. We hypothesize that the imbibition dynamics can be described by a combined model of the Lucas–Washburn equation and the Cox-Voinov law considering velocity-dependent contact angles. MethodsMolecular dynamics simulations are utilized to investigate the imbibition dynamics. A wide range of wetting conditions is achieved via adjusting the liquid–solid interaction parameters, and the spontaneous imbibition processes are quantified and compared. FindingsThe critical condition for the occurrence of spontaneous imbibition is analyzed from a surface energy perspective. The analyses of energy conversion and dissipation indicate that the viscous dissipation is dominant during spontaneous imbibition. The classical Lucas-Washburn equation is modified with the Cox–Voinov law considering the effect of the dynamic contact angle and an effective equilibrium contact angle. We show that the proposed theory well captures the imbibition dynamics embodied in the growth of imbibition length as well as the transient interface shape and velocity for both the symmetric and asymmetric wetting conditions. In nanochannels with asymmetric wettability, the imbibition length difference between the sidewalls and interface oscillations increases with wetting disparity. Our findings deepen the understanding of imbibition dynamics on the nanoscale, and provide a theoretical reference for relevant applications.

Spontaneous Imbibition in a Fractal Network Model with Different Wettabilities

In this work, we derived a mathematical model for spontaneous imbibition in a Y-shaped branching network model. The classic Lucas–Washburn equation was used for modeling the imbibition process occurring in the Y-shape model. Then, a mathematical model for the Newtonian fluid’s imbibition was derived to reveal the relationship between dimensionless imbibition time and length ratio, radius ratio, and wetting strength. The dimensionless imbibition time in the model was adopted to compare with that of the capillary bundle model. Different length and radius ratios were considered in the adjacent two-stage channels, and different wettabilities were considered in the different branches. The optimal radius ratio, length ratio, and wetting strength were calculated under the condition of the shortest imbibition time. In addition, the shortest dimensionless imbibition time of the three-stage Y-shaped branching network model was calculated when the wettability changes randomly. The results indicate that the imbibition time changed mostly when the wettability of the second branch changed, and the second branch was the most sensitive to wettability in the model.

Liquid imbibition into 3D printed porous substrates

Wettability analysis for granular material has been relying on the Washburn theory that assumes a particle bed as an array of parallel capillaries. However, the complexity in real-life powder beds or other industrial porous media usually are not fully taken into account. In the current study, we use three different porous substrates building from simple to complex as artificial models to represent porous particle beds for studying liquid imbibition. High-resolution Polyjet 3D printing technique was used to produce multiple copies for repetitive experimental study. The surface wettability was also tuned from hydrophilic to hydrophobic. The results of wetting analysis demonstrate that the porous structure plays a more significant effect on the liquid capillary rise than the material surface wettability. These experimental results also deviate from theoretical predictions using drop penetration time and Washburn equations, but pave the way for new production method for reproducible models of irregular powder beds.

Modelling of a Capillary Rise Height of Biochar by Modified Lucas–Washburn Equation

Modelling of a Capillary Rise Height of Biochar by Modified Lucas–Washburn Equation

Development of porous refractory ceramic plugs: Will the next generation be 3D printed?

Controlling non-metallic inclusion content during secondary metallurgy is a key feature in ensuring the quality of steel. In a typical process, these particles are captured by bubbles generated by a ceramic porous plug located at the bottom of the ladle, whose structure consists of open pores that allow gas injection into the molten bath. Nevertheless, during the non-bubbling period, liquid steel can penetrate through this porous structure, solidifying at the colder plug regions. In order to withdraw this infiltrated thickness, an oxygen lance heats the surface of the porous plug to resume its operation for the next process run. This cleaning procedure is pointed out as the leading wear mechanism for such a refractory. A classical approach to understand the infiltration of a liquid in a porous structure is based on Washburn's equation, which defines the starting condition for liquid penetration. By applying the same fundamentals to this problem, a criterion to avoid liquid steel infiltration into the plug is set. In this study, the influence of the pore diameter, the material composition, the gas counter-pressure and the ceramic structure tortuosity on steel penetration was analysed. Based on these results, feasible routes were proposed as countermeasures to refrain the molten metal infiltration of porous refractory ceramic purging plugs.

Simulating the molecular density distribution during multi-phase fluid intrusion in heterogeneous media

A computational recovery of multi-phase intrusion was discussed with the modified multi-relaxation time Lattice Boltzmann method (MRT-LBM). Originally proposed dual-matrix computation is developed to address the different phase separation and interface tracking for the multi-phase problem. A comprehensive validation is performed with the previously theorized observation of the mercury-water system. Results show that the dual-matrix computation is feasible to provide converged output under narrowed density difference down to 18%. The wetting and non-wetting behaviour resulted from form solid–fluid interaction is realized with arbitrary boundaries, in which the contact variance is up to 4.14%. The linear relations described by Laplace's law and Washburn's equation were three-dimensionally recovered with determination coefficients of 96.34% and 94.19%, respectively. A third fluid intrusion status of partial-intrusion is captured in addition to complete-intrusion and non-occupation in porous boundary, demonstrating the advanced function of the phase-separation and interface tracking in problems with further increased heterogeneity.

Spontaneous Imbibition Dynamics of Liquids in Partially-Wet Nanoporous Media: Experiment and Theory

Nanoscale spontaneous imbibition is a common process in nanoporous soil and unconventional reservoirs. Due to the complexity of these natural nanoporous media, the relevant spontaneous imbibition dynamics are still unclear. Thus, this paper studies spontaneous imbibition dynamics of liquids into a nanoporous carbon scaffold (NCS, with controllable wettability and pore geometry). The effects of evaporation and surfactant on spontaneous imbibition are also examined. For low-volatility liquids, a linear relationship between spontaneous imbibition height ( $$H$$ ) and square root of time ( $$\sqrt{t}$$ ) is observed. A modified Lucas–Washburn (L–W) equation is developed to describe the corresponding dynamics and to predict the NCS effective radius and the advancing water contact angle. A dimensionless time function is presented, which includes the properties of solid and liquid and their interactions, and can be used for upscaling spontaneous imbibition data in nanoporous media from laboratory to reservoir scales. Both a larger NCS pore diameter and pore throat diameter cause faster imbibition. The addition of surfactant into an aqueous solution increases the imbibition rate, although this influence is suppressed with pore size decrease. In contrast, for high-volatility liquids, a significant deviation from the linear relationship between $$H$$ and $$\sqrt{t}$$ is found at late times. The modified L–W equation is further developed to include evaporation effect. This study thus provides fundamental understanding of spontaneous imbibition dynamics at nanoscales. The developed theoretical models are expected to be applicable to important problems such as water infiltration into soil, fracturing fluid loss in unconventional reservoirs, and electrolyte migration in electrochemical devices.

Hemp-Based Microfluidics.

Hemp is a sustainable, recyclable, and high-yield annual crop that can be used to produce textiles, plastics, composites, concrete, fibers, biofuels, bionutrients, and paper. The integration of microfluidic paper-based analytical devices (µPADs) with hemp paper can improve the environmental friendliness and high-throughputness of µPADs. However, there is a lack of sufficient scientific studies exploring the functionality, pros, and cons of hemp as a substrate for µPADs. Herein, we used a desktop pen plotter and commercial markers to pattern hydrophobic barriers on hemp paper, in a single step, in order to characterize the ability of markers to form water-resistant patterns on hemp. In addition, since a higher resolution results in densely packed, cost-effective devices with a minimized need for costly reagents, we examined the smallest and thinnest water-resistant patterns plottable on hemp-based papers. Furthermore, the wicking speed and distance of fluids with different viscosities on Whatman No. 1 and hemp papers were compared. Additionally, the wettability of hemp and Whatman grade 1 paper was compared by measuring their contact angles. Besides, the effects of various channel sizes, as well as the number of branches, on the wicking distance of the channeled hemp paper was studied. The governing equations for the wicking distance on channels with laser-cut and hydrophobic side boundaries are presented and were evaluated with our experimental data, elucidating the applicability of the modified Washburn equation for modeling the wicking distance of fluids on hemp paper-based microfluidic devices. Finally, we validated hemp paper as a substrate for the detection and analysis of the potassium concentration in artificial urine.

Subpore‐Scale Trapping Mechanisms Following Imbibition: A Microfluidics Investigation of Surface Roughness Effects

AbstractSurface roughness is ubiquitous in subsurface formations due to weathering and diagenesis. Yet micromodel studies of porous media have not sufficiently addressed the impact of pore‐wall roughness on multiphase fluid displacement. We investigate the influence of pore‐wall roughness on capillary trapping by implementing surface roughness into glass micromodels and conducting imbibition experiments. Time‐lapse fluorescence microscopy, micromolding in capillaries, and scanning electron microscopy are used to examine flow behavior during spontaneous and forced imbibition experiments with capillary numbers between 10−6 and 10−4. It is found that surface roughness causes interface pinning and irregular displacement fronts within a single pore, promotes latent pore filling following imbibition, enhances snap‐offs, and contributes to interface relaxation. Furthermore, we verify a cooperative effect between corner flow and surface roughness that causes pore filling to occur at over 10 times the channel width ahead of the main bulk front. Consequently, a significant increase in trapped air saturation of up to 60% is observed, especially at lower capillary numbers and narrower pores. Effects of surface roughness also decrease the rate of spontaneous imbibition, which is slower than the prediction from Washburn's equation. Our results emphasize the importance of surface roughness in building conceptual models for capillary trapping when the ratio of surface roughness‐to‐pore size reaches a critical value of about 0.1. Above this critical threshold value, subpore‐scale trapping mechanisms contribute significantly to capillary trapping, which theoretical and numerical models should consider for accurate quantification of fluid‐transport processes.