Article pubs.acs.org/Langmuir
Effective Fluid Front of the Moving Meniscus in Capillary Chen Chen, Kangjie Lu, Lin Zhuang,* Xuefeng Li, Jinfeng Dong, and Juntao Lu College of Chemistry and Molecular Sciences, Wuhan University, Wuhan 430072, China S Supporting Information *
ABSTRACT: Traditionally, the meniscus bottom is taken as the fluid front when tracking the fluid motion in capillary, but in simulation studies, the thus-calculated motion curve deviates notably from the modified Lucas−Washburn equation. Here, we report that, by considering a volume equivalent of the meniscus part, the motion of the equivalent front agrees very well with the theoretical prediction; furthermore, such an effective fluid front can be directly represented by a specific position of the meniscus, which is independent of the capillary radius. These findings provide an accurate and practical method for describing the motion of the fluid front in capillary.
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INTRODUCTION The study of fluid confined in micropores is a foundation for a variety of applications, such as groundwater remediation, oil recovery, and nanofluidic devices.1,2 In this subject, computer simulations (e.g., molecular dynamics (MD), computational fluid dynamics (CFD), Lattice−Boltzmann method (LBM), dissipative particle dynamics (DPD), and its many-body extension (MDPD)) have been playing a significant role,3−7 not only complementary to the experimental observation, but necessary for in-depth understanding of the dynamic process in microscopic scale8−10 (in particular, in submicrometer and nanometer scale, where experimental observation is extremely difficult to pursue). In the present work, we deal with a very fundamental problem in the simulation of fluids in capillary, that is, how to properly describe the meniscus-shaped fluid front in motion. In textbooks, the position of the fluid front in a capillary is usually defined by the arc bottom of the meniscus.11 Such a practice does not cause any trouble in experiments, because the length of the fluid cylinder is usually much larger than the radius of the capillary, and the beginning stage of the fluid intake is often neglected. In computer simulations, however, the size of the system is relatively small, and the capillary radius can be comparable to its length; thus the accurate identification of the fluid front becomes critical, in particular at the beginning stage of the dynamic process. In this work, we take the spontaneous capillary imbibition (SCI) as an example to elucidate this problem. For the sake of simplicity, we only look at the completely wetting fluid in the present work, wherein the slip effect at liquid/solid interface can be neglected.12 In a simulation, as illustrated in Figure 1a and b, there are generally three choices of the fluid front at the meniscus: (i) the outmost layer,13,14 (ii) the central point,15,16 and (iii) somewhere between.17 The outmost layer of the fluid cylinder interacts intimately with the capillary wall, and its motion deviates from the bulk behavior. © XXXX American Chemical Society
Figure 1. Problems in the simulation study of the fluid-front motion in capillary. (a) Side view of the simulation system. (b) Axial view of the simulation system. (c) Representative motion curves resulted from different fluid-front identifications.
Therefore, this layer is usually not taken into account for both the contact angle fitting and the fluid motion monitoring. Some researchers tried to describe the motion of the fluid front based on the change in the fluid particle number18,19 or the density profile,20,21 but the accuracy was also affected by the abnormal Received: August 14, 2012 Revised: February 8, 2013
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fluid in capillary at each moment is important. We thus consider that the real front of the meniscus may be described by the edge of a volume equivalent of the meniscus part. Figure 2 illustrates a fluid in capillary with a curved interface (Figure 2a) and its volume-equivalent cylinder (Figure 2b).
behavior of the outmost layer. The central point of the meniscus is the arc bottom and has been assigned as the fluid front to describe the SCI motion. However, the motion of the central point of the meniscus appears not smooth in simulations, resulting in fluctuations in the motion curve and uncertainty in contact angle calculations. To avoid this problem, Martic et al.17 chose, somewhat arbitrarily, a layer of the fluid cylinder at 1/3 of the radius from the wall as the position of the fluid front. We are going to show that such a choice is practically correct and can be rationalized in view of the effective volume front.
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THEORY The motion of the fluid front can be considered as an average over the whole meniscus, with the outmost layer and the central point of the meniscus representing two extreme cases. As illustrated in Figure 1c, the motion curves (position versus time) of the outmost layer and the central point of the meniscus are rather different at the beginning stage of the SCI process. When a SCI process starts from a static contact angle of 0°, the central point of the meniscus seems to be dragged by a force greater than the average capillary force, and moves forward rapidly; the outmost layer, however, moves backward at the beginning and then forward at the same pace as the fluid bulk. Apart from these two extreme cases, there should exist an effective layer in the fluid cylinder, whose motion is driven by the average capillary force and fits the modified Lucas− Washburn equation,16 which describes the balance of the pressure on the fluid in capillary: 2γlv r
cos(θd) =
8η dl 1 d p(t ) + 2 lc c r dt π r 2 dt
Figure 2. The idea of using a volume equivalent of the meniscus part to determine the effective fluid front. (a) Fluid with a meniscus interface. (b) The volume equivalent of the fluid, whose edge is flat and well-defined.
Clearly, taking into account the volume of the meniscus part, the fluid length (lc) has to be extended. As shown below, by such a correction in lc, the motion curve of the fluid, deduced from simulation results, now agrees fully with the modified Lucas−Washburn equation. Theoretically, it is possible to obtain lc based on the instantaneous volume or particle number of the fluid in capillary. However, the volume of the fluid front is inconvenient to calculate at each time step in every simulation. In the case of completely wetting, the extending prewetting film at the capillary wall makes it difficult for meniscus determination. Although in numerical simulations, it is easy to count the particle number, but converting the number of liquid particles (or liquid mass) into the volume requires the liquid density to be precisely determined in advance, which, however, is affected by the wetting property of the fluid. In case of partial wetting, the density right at the fluid/capillary interface deviates from that of the bulk; vacuum regions can even be created between the capillary wall and the moving fluid in case of partly nonwetting. Our task is thus to find out if the front of the volume equivalent can be directly determined without calculating the volume of the meniscus part in practical simulations. To achieve this, we numerically calculate a number of regularly shaped menisci with different contact angles (θ), and then calculate the volume of the meniscus cap (V) and determine the volume front (the height of the volume-equivalent cylinder, hE, see Figure 3) through eq 3:
(1)
where γlv is the fluid surface tension, r is the radius of capillary, θd is the dynamic contact angle, η is the viscosity, lc is the fluid length in capillary, and p(t) is the momentum at time t (the explicit expression for dp(t)/dt is shown in the Supporting Information). Although the central point of the meniscus has traditionally been taken as the fluid front, its motion cannot be described by eq 1, in particular at the initial stage of the SCI process. To address this problem, Cupelli et al.22 introduced a parameter α to reduce the contribution of the fluid in the reservoir to the total momentum, such that the motion of the central point can be described by eq 1, but the choice of α depends on the size of the reservoir and also lacks of physical reasoning. We have been considering that this problem may be rationally addressed by a correct choice of the fluid front, rather than directly using the meniscus bottom. The correction in the fluid front will actually change the fluid length in capillary (lc), and consequently alter the motion curve calculated using the modified Lucas−Washburn equation (eq 1). The original Lucas−Washburn equation23,24 is known to be based on the famous Poiseuille’s law,25 which describes the pressure drop in a tube at a certain volumetric flow rate of fluid: ΔP =
8μLQ πr 4
(2)
where ΔP is the pressure drop along the capillary, μ is the fluid viscosity, L is the fluid length, Q is the volumetric flow rate, and r is the capillary pore radius. Apparently, ΔP is strongly related to the volume of the passing fluid within a certain amount of time dV/dt. Therefore, determining correctly the volume of
hEconcave
=
∭ dV πr 2
2
=
2
πr 2R + 3 π ( R2 − r 2 )3 − 3 πR3 πr 2 (3a)
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Figure 3. Illustration for the calculation of volume-front point (eqs 3 and 4). Parts (a) and (b) are the side views of the concave and convex meniscus, respectively, and (c) is the axial view.
hEconvex = =
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RESULTS AND DISCUSSION To examine the above method, a series of many-body dissipative particle dynamics (MDPD) simulations were carried out with different capillary radii (5, 10, 15, 20 in MDPD length unit). MDPD simulations were performed using the LAMMPS package26 by modifying the original DPD codes therein. A brief description of MDPD, as well as computational details, can be found in the Supporting Information. Simulation results are summarized in Figure 5, in comparison to the theoretical
∭ dV πr 2 2 πR3 3
2
1
− 3 πR2 R2 − r 2 − 3 πr 2 R2 − r 2 πr 2 (3b)
where symbols are as defined in Figure 3. The intersection of the meniscus and the volume front is defined as the volumefront point, whose radial coordinate (rv) satisfies the following equation: (rvconcave)2 + (R − hEconcave)2 = R2
(4a)
(rvconvex )2 + (R sin θ + hEconvex )2 = R2
(4b)
Upon analyzing all of the simulated menisci, we found that the ratio rv/r, called the relative radial coordinate of the volumefront point, is approximately independent of the contact angle. As demonstrated in Figure 4, in both wetting and nonwetting cases (concave and convex meniscus, respectively), rv/r changes only slightly from 0.707 to 0.745 with a mean value of 0.721 and a standard deviation of 0.013. Therefore, in practice, one can directly take the position of the meniscus at 72.1% of the capillary radius from the center as the effective fluid front.
Figure 5. MDPD simulation results of the motion curve of the effective fluid front in capillaries with different radii (5, 10, 15, 20 in MDPD length unit), all in excellent agreement with the modified Lucas−Washburn equation (eq 1).
predictions using eq 1. Strikingly, all of the motion curves derived from simulation data are in excellent agreement with the theoretical curves, clearly proving that taking the position of the meniscus at 72.1% of the capillary radius as the effective fluid front is a proper choice to describe the fluid motion in capillary. To quantitatively demonstrate the superiority of our method, we compare the deviations between the simulated motion curve and the prediction of Lucas−Washburn equation for three typical choices of the position at the meniscus. As demonstrated in Figure 6, for all of the capillary radii here simulated, the deviations are not ignorable when using either the central point or the outmost layer of the meniscus as the fluid front. Whereas the central point tends to move faster at the initial period and then slower afterward, the outmost layer usually moves at a slower speed initially and faster afterward. As the radius of capillary increases, both the magnitude of the deviation and the duration of the meniscus relaxation increase.
Figure 4. The relative radial coordinate (rv/r) of the volume-front point at the meniscus turns out to be approximately independent of the contact angle (θ). Both the wetting and the nonwetting case (concave and convex meniscus, respectively) give the same result. C
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Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was financially supported by the National Basic Research Program (2012CB215503, 2012CB932800), the National Science Foundation of China (20933004, 21125312), the Doctoral Fund of the Ministry of Education of China (20110141130002), the Program for Changjiang Scholars and Innovative Research Team in University (IRT1030), and the PetroChina Changqing Oilfield Co.
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(1) Alava, M.; Dube, M.; Rost, M. Imbibition in Disordered Media. Adv. Phys. 2004, 53, 83−175. (2) De Gennes, P. G. Wetting: Statics and Dynamics. Rev. Mod. Phys. 1985, 57, 827−863. (3) Gruener, S.; Sadjadi, Z.; Hermes, H. E.; Kityk, A. V.; Knorr, K.; Egelhaaf, S. U.; Rieger, H.; Huber, P. Anomalous Front Broadening during Spontaneous Imbibition in a Matrix with Elongated Pores. Proc. Natl. Acad. Sci. U.S.A. 2012, 109, 10245−10250. (4) Thomas, J. A.; McGaughey, J. H. Water Flow in Carbon Nanotubes: Transition to Subcontinuum Transport. Phys. Rev. Lett. 2009, 102, 184502. (5) Chen, C.; Zhuang, L.; Li, X.; Dong, J.; Lu, J. A Many-Body Dissipative Particle Dynamics Study of Forced Water−Oil Displacement in Capillary. Langmuir 2012, 28, 1330−1336. (6) Stukan, M. R.; Ligneul, P.; Crawshaw, J. P.; Boek, E. S. Spontaneous Imbibition in Nanopores of Different Roughness and Wettability. Langmuir 2010, 26, 13342−13352. (7) Leoni, F.; Kierlik, E.; Rosinberg, M. L.; Tarjus, G. Spontaneous Imbibition in Disordered Porous Solids: A Theoretical Study of Helium in Silica Aerogels. Langmuir 2011, 27, 8160−8170. (8) Blake, T. D. The Physics of Moving Wetting Lines. J. Colloid Interface Sci. 2006, 299, 1−13. (9) De Coninck, J.; Blake, T. D. Wetting and Molecular Dynamics Simulations of Simple Liquids. Annu. Rev. Mater. Res. 2008, 38, 1−22. (10) Bonn, D.; Eggers, J.; Indekeu, J.; Meunier, J.; Rolley, E. Wetting and Spreading. Rev. Mod. Phys. 2009, 81, 739−805. (11) Atkins, P.; de Paula, J. Atkins’ Physical Chemistry, 8th ed.; Oxford University Press: Oxford, 2006. (12) Neto, C.; Evans, D. R.; Bonaccurso, E.; Butt, H.; Craig, V. S. J. Boundary Slip in Newtonian Liquids: A Review of Experimental Studies. Rep. Prog. Phys. 2005, 68, 2859−2897. (13) Ahadian, S.; Kawazoe, Y. A Molecular Dynamics Approach to Examine the Kinetics of the Capillary Imbibition of a Polymer at Nanoscale. Colloid Polym. Sci. 2009, 287, 961−967. (14) Chibbaro, S. Capillary Filling with Pseudo-Potential Binary Lattice-Boltzmann Model. Eur. Phys. J. E 2008, 27, 99−106. (15) Chen, C.; Gao, C. N.; Zhuang, L.; Li, X. F.; Wu, P. C.; Dong, J. F.; Lu, J. T. A Many-Body Dissipative Particle Dynamics Study of Spontaneous Capillary Imbibition and Drainage. Langmuir 2010, 26, 9533−9538. (16) Cupelli, C.; Henrich, B.; Glatzel, T.; Zengerle, R.; Moseler, M.; Santer, M. Dynamic Capillary Wetting Studied with Dissipative Particle Dynamics. New J. Phys. 2008, 10, 043009. (17) Martic, G.; Gentner, F.; Seveno, D.; Coulon, D.; De Coninck, J.; Blake, T. D. A Molecular Dynamics Simulation of Capillary Imbibition. Langmuir 2002, 18, 7971−7976. (18) Kutana, A.; Giapis, K. P. Atomistic Simulations of Electrowetting in Carbon Nanotubes. Nano Lett. 2006, 6, 656−661. (19) Supple, S.; Quirke, N. Rapid Imbibition of Fluids in Carbon Nanotubes. Phys. Rev. Lett. 2003, 90, 214501. (20) Gruener, S.; Hofmann, T.; Wallacher, D.; Kityk, A. V.; Huber, P. Capillary Rise of Water in Hydrophilic Nanopores. Phys. Rev. E 2009, 79, 067301.
Figure 6. The deviation between the simulated motion curve and the prediction from the modified Lucas−Washburn equation (eq 1). Different capillary radii are involved, and different choices of the position at the meniscus are compared (OL, outmost layer; VF, volume front; CP, central point).
On the contrary, when using the volume-front point for motion monitoring, a good consistency between the simulation result and the theoretical curve can be achieved for capillary with different radii. As mentioned in the Introduction, Martic et al.17 took a position of the meniscus at 2/3 of capillary radius as the fluid front. The difference between the choice of Martic et al. and ours (0.721 of the radius to the center) is not just a seemingly slight variation in number, but rather the rationality. In contrary to the arbitrary choice they made, ours has a rational and clear physical foundation. Although it is still an open question whether Poiseuille’s law and the Lucas−Washburn equation, known to be theories for tube of infinite length, can be applicable to confined fluid with finite length,27,28 our results, as well as many previous simulation studies, have shown that, at least to a good approximation, these principles are still practical, even for the initial stage of capillary imbibition. A possible reason we can figure for such a seemingly “unreasonable” success could be the use of dynamic contact angle (θd), rather than the static contact angle (θ0), in solving the differential equation of pressure (eq 1). The effect of the pressure change in the simulated imbibition process could have been well corrected in θd. Our findings here reported have strengthened the basis of the simulation study of fluid confined in micropores, and also furthered our fundamental understandings of this subject. Although the volume-front point is deduced on the basis of model capillary with smooth wall, it should also be applicable to the experimental studies29,30 using well-shaped capillaries, where a fixed position at the meniscus should not be difficult to monitor in comparison to the meniscus bottom.
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ASSOCIATED CONTENT
S Supporting Information *
Simulation details. This material is available free of charge via the Internet at http://pubs.acs.org.
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REFERENCES
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. D
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