Lucas-Washburn Equation Research Articles

Effect of gradient wettability on capillary imbibition in open semicircular copper channel

Open semicircular copper channels with a gradient wettability of 45°–3° or 3°–45° (contact angle of 1,2-propanediol), homogeneous wettability of 3°, and segmented wettability on their inner surfaces were constructed. The capillary imbibition of 1,2-propanediol in these channels demonstrates that, compared with the surface with homogeneous wettability, there are an additional driving force and resistance on the surfaces with a gradient wettability of 45°–3° and 3°–45°, respectively. Meanwhile, the channel with gradient wettability on the first half segment and homogeneous wettability on the second half segment could accelerate capillary imbibition better than that with gradient wettability on the whole channel. Furthermore, a smaller length ratio, i.e., length of gradient segment to that of homogeneous segment of the channel, demonstrates a better liquid acceleration effect. Based on the Lucas–Washburn equation, a theoretical model to describe capillary imbibition was established by introducing an additional force arisen from the positive gradient wettability and a retardation coefficient caused by the surface roughness. The flow front position vs flow time could be predicted by the established model in this work, and the relative error between the theoretical prediction and experimental result was less than 20.7%, indicating the rationality of the proposed model.

Polyethylene three-dimensional nano-networks: How lateral chains affect metamaterial formation

Polymers occupy a central role in current society due to their utility and versatile properties. As part of the class of soft materials, polymers have also been employed in research through nanostructuration and functionalization, looking for the development of new metamaterials with wide applicability in science and society. A particular polymer, Polyethylene, has been widely used both in research and commodities, and new ways to nanostructure and improve its functionality should be considered. Here, we report on the nanostructuration of polyethylene for the fabrication of three-dimensional nanonetworks, by resorting to 3 different variants of polyethylene: low density, high density, and ultra-high molecular weight. Through the melt infiltration of these polymers into well-known three-dimensional interconnected anodic aluminum oxide templates (3D-AAO), a study can be performed to understand the effects of the chain length, structure, and chain-branching of the used polymers on the stability and integrity of the resulting polymeric nanonetworks. This was accomplished by performing infiltrations in the presence of excess bulk material and quenching the infiltration process to access transient infiltration stages. The morphology of these networks was analyzed through SEM and STEM-HAADF to understand the differences arising from polymer structure. The observed results are interpreted through the use of the Lucas-Washburn equation for capillary flow and the determination of the critical contact angle for spontaneous capillary infiltration. The resulting metamaterials also exhibit photonic responses, resulting from the replication of the periodic nature of the employed templates.

Water distribution characteristics and research with capillary absorption for magnesium phosphate cement-coated cement pastes

Magnesium phosphate cement (MPC) is a cementitious material formed by an acid-base reaction, which has usually developed as repair material for rehabilitation engineering. It also has the potential as coating materials for fire-resistance and for corrosion protection. In this study, the effect of magnesium phosphate cement as an inorganic coating on the water permeability of cement-based material is studied. X-ray computed tomography (XCT) was used to track the capillary adsorption process of hardened cement non-destructively. The results showed that magnesium phosphate cement as the coating of cement-based materials has an excellent resistance to water penetration, and the quantitative and qualitative analysis of XCT water penetration was conducted. The microstructure analysis showed that the MPC matrix has much smaller porosity and pore sizes than the OPC matrix, and the interphase of OPC and MPC is dense and continuous, which may provide a remarkable effect on the capillary absorption blocking process. According to the Lucas-Washburn equation, a linear relationship between water absorption height and the square root of time was established to verify the experimental results measured/calculated by XCT.

Rate of capillary rise in quartz nanochannels considering the dynamic contact angle by using molecular dynamics

As a universal phenomenon in nature, capillary rise not only strongly influences the physical properties and industrial processes of powder materials but can also be applied as an important technique to measure their wettability. In this paper, molecular dynamics (MD) was implemented to investigate the capillary transport behavior of water flow through α-quartz nanochannels considering the channel height. The results showed that the Capillary rise method (CRM) with the classical Lucas-Washburn (L-W) equation may obtain an overestimated equilibrium contact angle, and the value decreases with increasing channel height. Through the introduction of the concepts of effective viscosity, slip length and dynamic contact angle, a modified L-W equation was proposed. The equilibrium contact angle calculated by the equation was more accurate and independent of channel size. Additionally, the pore-water can be divided into absorption water and capillary water. The upper bound of the absorption water density was determined to be 1.14 g/cm3.

Capillary imbibition of confined water in nanopores

The spontaneous capillary imbibition of confined nanopores is investigated using an analytical model that includes the slip effect, wettability and effective viscosity at the water surface interface. The results show that the effective viscosity of confined water is larger than that of bulk water and decreases with diameter and wettability. The effective slip length is negative for a contact angle of 0◦, and the effective slip length is positive and increases with diameter. The results of the presented model show that the capillary imbibition length for nanoconfined water can vary up to 0.389-1.033 times that determined by the Lucas-Washburn equation with no-slip boundary conditions for nanopores due to the effective viscosity and slippage with various dimensions and contact angles. The enhancement increases with diameter and contact angle. These results elucidate the confined movement through nanopores, which can be used to understand fracturing-fluid flow in the nanopores of shale reservoir formations. Cited as : Zeng, F., Zhang, Q., Guo, J., Meng, Y., Shao, X., Zheng, Y. Capillary imbibition of confined water in nanopores. Capillarity, 2020, 3(1): 8-15, doi: 10.26804/capi.2020.01.02

Inverse modeling deduction of pore distribution in cement materials from capillary absorption features

The moisture transport performance of concrete is a vital factor that affects its durability, and this performance is closely related to the pore structures of the porous materials. In this study, water content profiles of capillary absorption were obtained using X-ray computed tomography technology. Based on the Lucas–Washburn equation, an inverse model was established to deduce the pore size distribution from the water content profiles. The feasibility of the proposed method was demonstrated using a size distribution model of the cement samples with different water-to-cement ratios. The obtained results were compared with the cumulative pore size distribution measured by mercury intrusion porosimetry, which indicated that the two methods exhibited satisfactory consistency.

A theoretical study on the capillary rise of non-Newtonian power-law fluids

The phenomena of capillary rise with the effect of displaced fluids are ubiquitous in both science and engineering, and the mathematical models and their analytical solutions of this problem have also received increasing attention. In this paper, a theoretical study on the rising dynamics of non-Newtonian power-law fluids in a capillary is performed, and the classical Lucas–Washburn equation is generalized to a nonlinear second-order differential equation in which the effects of the power-law index and displaced fluids are included. We analyze the imbibition behaviors of power-law fluids under the influence of displaced fluids in details, and also present some analytical solutions of different special cases and different time stages. The results show that for different special cases, it takes shorter time for shear thickening fluid to reach the equilibrium height. For different time stages, however, the rising phenomena of power-law fluids are more complex, and the average time for the shear-thinning fluid to reach the equilibrium height is longer compared to shear-thickening fluid, but an opposite phenomenon is observed for the case of μ1,0/μ2,0=100 (here μ1,0/μ2,0 is the viscosity ratio of two power-law fluids) in viscous time stage. In inertial time stages, the density ratio of the absorbed fluid to the displaced fluid also has a significant effect on the rising dynamics of imbibing fluid. Furthermore, the effect of dynamic contact angle is also included in the governing equation and analytical solutions. Through a comparison between the theoretical results and experimental data, a good agreement is observed. These results can be used as a priori for liquid absorption in industrial applications, including oil recovery, fuel cells and water collection of artificial silk.

Solution based freeze cast polymer derived ceramics for isothermal wicking - relationship between pore structure and imbibition

ABSTRACT Besides conventional applications for porous monoliths such as filtration, open porosity enables capillary transport. For this application, porous polymer-derived ceramic monoliths (SiOC) with different pore morphology and porosity were obtained by solution-based freeze casting. Methylpolysiloxane and (3-aminopropyl)triethoxysilane were used as a precursor and cross-linking agent, respectively. Tert-butyl alcohol and cyclohexane acted as solvents and created prismatic and dendritic pore morphology. Additionally, differences in solid loading and the addition of preceramic filler particles changed the open porosity from 62% to 79%, the mean pore window diameter from 11 µm to 21 µm and the isotropy. The lateral surface of the monoliths is mainly closed due to the use of a Si-coated film as an intermediate layer which prevents nucleation. Within the parameters characterizing the pore structure, open porosity and pore window diameter were found to be most influencing on wicking tested by the Washburn-Sorption method. The permeability was obtained by constant head experiments and from the viscous-dominated part of the wicking curve. Furthermore, predictions of wicking using the Lucas–Washburn equation with gravity effect which bases on the assumption of capillary bundles were conducted. Wicking experiments showed that describing a real porous structure by macroscopic parameters may not be sufficient for structures deviating strongly from the assumption of capillary bundles. The combination of prediction with the knowledge of main influencing factors allows for tailoring the pore structure of SiOC monoliths prepared by solution-based freeze casting for capillary transport applications.

Unidirectional solution-based freeze cast polymer-derived ceramics: influence of freezing conditions and templating solvent on capillary transport in isothermal wicking

Porous SiOC monoliths were prepared by solution-based freeze casting of polysiloxane at constant freezing temperature or constant freezing front velocity. Dendritic and prismatic pore structures were obtained by using cyclohexane and tert-butyl alcohol as solvent, respectively. Gradients in freezing velocity lead to gradients in pore window size, whereas a constant freezing velocity (3.3–6.8 µm/s) generates homogeneous pore structures. The water permeability varies from 1.12 × 10−13 to 1.03 × 10−11 m2 and correlates with the pore window diameter (10–59 µm) and the porosity (51–82%). In wicking tests, the gradient in pore window size is clearly reflected by a pronounced decrease in the wicking speed. Contrary, a homogeneous pore structure results in wicking curves which are closer to the prediction according to the Lucas–Washburn equation. However, this theoretical approach based on the three parameters, pore window size, porosity and permeability, is insufficient to describe complex three-dimensional pore structures. Besides the porosity, the pore morphology was found to be a major influencing factor on the wicking. The filling of secondary dendrites slows down the wicking into the dendritic structure. Fastest wicking was observed for a prismatic pore structure at low freezing front velocity (6.6 µm/s) and high porosity (78%), whereas slowest wicking occurred into the dendritic structure with high porosity (76%) and constant freezing temperature (− 20 °C). The knowledge of the relationship between structural properties and the resulting wicking behavior can address a variety of pivotal applications in chemical engineering for capillary transport.

Multilayered Microfluidic Paper-Based Devices: Characterization, Modeling, and Perspectives.

Microfluidic paper-based analytical devices (μPADs) are simple but powerful analytical tools that are gaining significant recent attention due to their many advantages over more traditional monitoring tools. These include being inexpensive, portable, pump-free, and having the ability to store reagents. One major limitation of these devices is slow flow rates, which are controlled by capillary action in the hydrophilic pores of cellulosic paper. Recent investigations have advanced the flow rates in μPADs through the generation of a gap or channel between two closely spaced paper sheets. This multilayered format has opened up μPADs to new applications and detection schemes, where large gap sizes (>300 μm) provide at least 169× faster flow rates than single-layer μPADs, but do not conform to established mathematical models for fluid transport in porous materials, such as the classic Lucas-Washburn equation. In the present study, experimental investigations and analytical modeling are applied to elucidate the driving forces behind the rapid flow rates in these devices. We investigate a range of hypotheses for the systems fluid dynamics and establish a theoretical model to predict the flow rate in multilayered μPADs that takes into account viscous dissipation within the paper. Device orientation, sample addition method, and the gap height are found to be critical concerns when modeling the imbibition in multilayered devices.

Capillary imbibition of ethanol in cement paste traced by X-ray computed tomography with CsCl-enhancing technique

This study reported the capillary imbibition of ethanol in a cement paste traced by X-ray computed tomography (X-CT) enhanced by CsCl. Results show that CsCl can promote the contrast of X-CT images due to the significant X-ray absorption of Cs. The grey value distributions of horizontal section images were calculated to identify the penetration fronts of the CsCl-ethanol solution in the cement paste. The relationship between imbibition height and time follows the Lucas-Washburn equation. The method of enhancing X-ray sorption by CsCl paves a wide path to in situ monitor liquid imbibition in porous materials for assessing their microstructures and properties.

Mathematical modeling of the dissolution of phosphate rock into various acidic medium

The leaching of phosphate ores in acid medium is widely practiced in the chemical fertilizers industry. Several processes can take place in the reactor and the quantitative modeling of the reaction kinetics or thermodynamics is essential for the design and dimensioning of the industrial equipments. The aim of the present study is to develop a mathematical model to describe the behavior of phosphate particles into acidic solution under certain conditions. A new formalism based on the Noyes-Whitney equation (NWE) is presented in this paper. In this approach, simplified Lucas-Washburn's equation (LWE) is incorporated to evaluate the progression of reaction zone into the mesoporous phosphate rock. The reliability and validity of this new approach are tested on a large database. Results show that the model is predictive in the case of a solid dissolution without ash film and non predictive in the case of solid leaching with formation of a rushed in the mesoporous area follows. Furthermore, the values and physical significance of the parameters of this model are also discussed in the present work.

Meniscus Motion and Void Generation Inside Carbon Nanotubes

The hollow inside of a carbon nanotube (CNT) has great potential not only for flow rate enhancement of nanocapillary, but also for a material container which can be applied for drug delivery and nanoparticle infusion. However, these applications focus on after liquid infusion into CNTs, whereas the understanding of the filling process is still limited. We conducted capillary filling experiments using individual open-ended CNTs, which were stuck into an ionic liquid and visualized by scanning transmission electron microscopy. The results showed that the meniscus stopped inside the CNT, which is not predicted by the Lucas–Washburn equation. To explain this discrepancy, the intermolecular force between the liquid and CNT inner wall was proposed to provide an additional friction force. In addition, voids were observed in the liquid inside the CNT. The generation mechanism of voids was proposed to be induced by the instability of the thin liquid layer along the CNT inner surface caused by the advance of the th...

Modeling of capillary-driven flows in axisymmetric geometries

We present an analytical approach, as well as computer simulations based on the free surface lattice-Boltzmann (FSLB) method, in order to model capillary-driven infiltration of liquids into porous structures. The analytical method is an extension of the Lucas-Washburn (LW) equation and applies to axisymmetric geometries with a circular cross-section. The treatment of irregular capillaries is achieved by a discretization procedure in which the original geometry is divided into small cylinders. In order to validate the derived analytical equation, we perform FSLB simulations in test geometries which show a good agreement.

Flow of liquids through paper

The flow of liquids through paper is challenging to model due to the complexity and disordered layout of the fibre matrix. The expanding use and capability of microfluidic paper-based analytical devices ($\unicode[STIX]{x1D707}$PADs) and their requirement for precision has increased the need to accurately predict the flow of liquids through paper. Many studies have developed models and revealed some of the physical mechanisms responsible for the flow behaviour, but we still lack a complete understanding, particularly in relation to how the fluid fills the various voids with a wide range of shapes and sizes in the fibre matrix of paper. In the featured article, Chang et al. (J. Fluid Mech., vol. 845, 2018, 36–50) used a combined experimental and theoretical approach to uncover the importance of the liquid filling the intra-fibre pores, showed that this results in deviation from the flow behaviour predicted by the Lucas–Washburn equation and developed a model which accounts for this effect.

Molecular Dynamics Study on the Structure and Dynamics of NaCl Solution Transport in the Nanometer Channel of CASH Gel

The transport of water molecules and ions in the nanopores of calcium aluminosilicate hydrate (CASH) influences the durability and sustainability of environmentally friendly cement-based materials with industrial waste substitution. In this study, molecular dynamics was utilized to study aqueous NaCl solution capillary transport through the calcium silicate hydrate (CSH) and calcium aluminosilicate hydrate (CASH) gel pore with pore size of 3.2 nm. The invading depth for the NaCl solution advancing frontier with meniscus shape follows a parabolic relation as a function of time, consistent with the classic Lucas–Washburn equation in capillary adsorption theory. As compared with the solution transport in the CSH pore, both water molecules and ions migrate more slowly in the gel pore of CASH, and sodium ions accumulate in the entrance region of the gel pore. The incorporation of Al atoms in the silicate substrate resists the ingress of ions and water. The Al–Si substitution on the CASH interface enhances the ...

Theory on Capillary Filling of Polymer Melts in Nanopores.

A unified theory for the imbibition dynamics of entangled polymer melting into nanopores is presented. Experiments demonstrate the validity of t1/2 dependence but contradict the predictions of the classical Lucas-Washburn equation because of the prefactor. A reversal in dynamics of capillary filling is reported with increasing polymer molecular weight. Polymer imbibition under nanometer confinement can be discussed by two mechanisms: one is the standard hydrodynamic flow, resulting in a parabolic flow profile. When the inner wall has a strong attraction to the polymer, a layer of immobile chains is created, resulting in an increase of the effective viscosity and to slower imbibition. The other is the reptation model proposed by Johner et al., leading to a plug flow profile and to the reduction in the effective viscosity (faster imbibition). The reversal in dynamics of polymer imbibition can be explained by the competition between these two mechanisms.

Capillary Imbibition of Polymer Mixtures in Nanopores

Capillary imbibition of homogeneous mixtures of entangled poly(ethylene oxide) melts in nanopores of self-ordered nanoporous alumina follows a t1/2 dependence but contradicts the classical Lucas–Washburn equation. Herein we employ reflection microscopy and self-consistent field theory (SCFT) calculations to demonstrate the faster penetration of nanopores for the shorter chains. Combined results suggest on average an ∼15% enrichment by the shorter chains. On top of that, SCFT shows an enrichment of the short chains near the pore surface. Possible applications in separating long and short polymer chains by the difference in imbibition speed—in the absence of solvent—are discussed.

Wetting boundary condition for the color-gradient lattice Boltzmann method: Validation with analytical and experimental data

Abstract In the color gradient lattice Boltzmann model (CG-LBM), a fictitious-density wetting boundary condition has been widely used because of its ease of implementation. However, as we show, this may lead to inaccurate results in some cases. In this paper, a new scheme for the wetting boundary condition is proposed which can handle complicated 3D geometries. The validity of our method for static problems is demonstrated by comparing the simulated results to analytical solutions in 2D and 3D geometries with curved boundaries. Then, capillary rise simulations are performed to study dynamic problems where the three-phase contact line moves. The results are compared to experimental results in the literature (Heshmati and Piri, 2014). If a constant contact angle is assumed, the simulations agree with the analytical solution based on the Lucas–Washburn equation. However, to match the experiments, we need to implement a dynamic contact angle that varies with the flow rate.

Characterization of adhesive interphase between epoxy and cement paste via Raman spectroscopy and mercury intrusion porosimetry

When exposed to harsh environmental conditions, the adhesive bond between epoxy and concrete was found to depend almost entirely on the effectiveness of the mechanical interlock which is contingent upon the ability of the adhesive to properly wet and penetrate the substrate. The current study is an investigation of the existence of a distinct interphase region within cement paste through the combined use of mercury intrusion porosimetry and Raman spectroscopy. A positive correlation between median pore size and depth of epoxy permeation into the cement paste substrate was observed. The Lucas-Washburn equation was found to accurately represent the driving mechanisms behind permeation of curing epoxy into the porous cement paste matrix. Interconnectivity of the capillary and gel pore structures was determined to play an important role in the permeation mechanisms. Analysis of relative conversion of epoxy via Raman spectroscopy in the interfacial region showed evidence of epoxy-cement paste interactions.