- Email: [email protected]
Free-surface dynamics of small pores
Chemical Engineering Science 132 (2015) 93–98
Contents lists available at ScienceDirect
Chemical Engineering Science journal homepage: www.elsevier.com/locate/ces
Free-surface dynamics of small pores Jiakai Lu, Carlos M. Corvalan n Transport Phenomena Laboratory, Department of Food Science, Purdue University, West Lafayette, IN 47907, USA
H I G H L I G H T S
We consider the contraction of small pores nucleated in fluid sheets. We characterize the pore dynamics solving the full Navier–Stokes system. A scaling function without free parameters describes the size of closing pores near collapse. Contracting pores may reverse the direction of motion (flicker) and expand.
art ic l e i nf o
a b s t r a c t
Article history: Received 17 January 2015 Received in revised form 28 March 2015 Accepted 4 April 2015 Available online 17 April 2015
When the size of a pore nucleated in a fluid sheet is sufficiently small, the pore will contract and close driven by its large radial curvature. The dynamics of small contracting pores are relevant to a number of natural systems and practical applications, from fission pores in cell membranes to the fabrication of nanopores in sensors for DNA sequencing. Here, we report high-fidelity numerical simulations that provide detailed insight into the mechanisms of pore contraction and collapse in fluid sheets of low viscosity. Results uncover a scaling law that predicts the radius of a closing pore as a function of the time to collapse without free parameters. Simulations also show that contracting pores do not always proceed to collapse. Instead, some contracting pores reverse the direction of motion and expand. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Nanopore Pore Interface Simulation Transport processes Cell membrane
1. Introduction This paper reports a numerical study of the free-surface dynamics of small pores in fluid sheets. Unlike large pores, sufficiently small pores contract and close driven by surface tension and their large radial curvature—a behavior discussed in the pioneering work by Taylor and Michael (1973). The dynamics of small contracting pores in fluid sheets impact a wide range of systems, from electroporation and fission pores in cell membranes to the stability of food foams (Moroz and Nelson, 1997; Wanunu, 2012; Zhao et al., 2010). Moreover, understanding and ultimately controlling the dynamics of small pores in the micro and nanoscales offer potential applications for the fabrication of sensors for rapid characterization of biomolecules and DNA sequencing (Storm et al., 2003, 2005; Dekker, 2007; Schneider and Dekker, 2012).
n
Corresponding author. E-mail address: [email protected] (C.M. Corvalan).
http://dx.doi.org/10.1016/j.ces.2015.04.013 0009-2509/& 2015 Elsevier Ltd. All rights reserved.
Although the dynamics of large expanding pores have been extensively studied and the speed of expansion is now well characterized (Culick, 1960; Taylor, 1959; Ranz, 1950; Savva and Bush, 2009; Debregeas et al., 1995, 1998; Keller et al., 1983; Keller and Mikiss, 1983), surprisingly little is known about the freesurface dynamics of small contracting pores and their speed of collapse. Indeed, in a recent theoretical and numerical work, Savva and Bush (2009) investigated the free-surface dynamics of circular pores using a lubrication model in the long wavelength limit. Their findings provided important insights into the early stage dynamics of pore expansion, and confirmed the exponential growth rate observed in experiments. However, due to limitations intrinsic to the lubrication approximation, the lubrication model cannot predict the dynamics of small contracting pores. To study the free-surface dynamics of small pores, we have developed high-fidelity simulations that overcome the limitations of the lubrication model by solving the full Navier–Stokes (NS) equations. Here, we report how the solution of the full NS system enables a detailed analysis of the free-surface dynamics of pore contraction (Section 3.2), and uncovers a new scaling law that predicts the size of collapsing pores near the singularity without
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free parameters (Section 3.3). Results also characterize the behavior of small contracting pores that suddenly change the direction of motion and expand (flicker) by purely hydrodynamic reasons (Section 3.4).
2. Problem description We now state our model and assumptions. We consider the free-surface dynamics of circular pores nucleated in a fluid sheet of density ρ, viscosity μ, and surface tension σ, as sketched in Fig. 1. The model and results are described in this paper using the thickness of the fluid sheet H as characteristic length scale, the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi inviscid time τ ρH 3 =σ as characteristic time scale, and the
stress μ=τ as characteristic pressure scale. To analyze the pore dynamics, we numerically solved the full axisymmetric Navier–Stokes and continuity equations: ∂v þ v ∇v ¼ ∇p þ ∇2 v; ð1Þ Re ∂t ∇ v ¼ 0;
ð2Þ
for the velocity v and pressure p. The pReynolds number ffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi Re ρσ H =μ based on the inviscid pffiffiffi velocity σ =ρH is related to the Ohnesorgepnumber ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Oh ¼ 1= 2 Re based on the Taylor–Culick velocity v^ c ¼ 2σ =ρH (Taylor, 1959; Culick, 1960). We further assume that gravity can be neglected compared with surface tension. Along the sheet interface both the traction boundary condition T n ¼ 2 Re Hn;
0.3 0.2 0.1
ð3Þ
0
and the kinematic boundary condition n ðv vs Þ ¼ 0
Fig. 1. Small circular pore on a fluid sheet with density ρ, viscosity μ, and surface tension σ. The thickness of the fluid sheet is H and the initial pore radius is R0.
ð4Þ
were imposed, where T is the stress tensor, vs the velocity of the points at the interface, and n the unit normal vector to the interface (Slattery et al., 1990). The interfacial curvature is H ¼ 1=2 ðκ z þ κ r Þ, where κz and κr are the axial and radial curvature, respectively (Slattery et al., 1990). In addition, symmetry boundary conditions were imposed on the plane z¼ 0 and axis of symmetry r ¼ 0. We used the finite-element method along with the arbitrary Lagrangian–Eulerian method of spines pioneered by Kistler and Scriven (1983) to parametrize the spatial derivatives and the deforming pore interface, following a design that we have successfully applied to similar free-surface flows (Xue et al., 2008; Muddu et al., 2012; Lu and Corvalan, 2012, 2014). The time was discretized using a second-order trapezoidal method with Adam–Bashforth prediction to reduce time truncation errors. To improve computational efficiency, the time steps were calculated using a first-order continuation method (Corvalan and Saita, 1991).
3. Results and discussion The qualitative behavior of a pore nucleated in a fluid sheet may change substantially with the initial pore size. There are at least four conceivable behaviors: the pore may open, close, flicker or remain stable. Here we discuss three of these scenarios, which are summarized in Fig. 2. The figure compares the time evolution of three different pores which open (long dashed line), close (dashed line), or flicker (solid line) depending solely upon the initial pore size R0. We have not observed stable pores (in the sense of Taylor and Michael, 1973) for the parameters considered in this paper.
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Fig. 2. Influence of initial size on the dynamics of pore formation. Time evolution of the minimum pore radius r0 in an inertial fluid sheet with Oh¼ 0.04. Depending on the initial radius R0, the pore: (a) expands (long-dashed line, R0 ¼ 50), (b) contracts (dashed line, R0 ¼ 0:2), or (c) initially contracts and then expands (solid line, R0 ¼ 0:36).
3.1. A large expanding pore Although our focus is on the little-known dynamics of small pores, we begin our discussion with the large opening pore introduced in Fig. 2a in order to make comparisons with a recent work by Savva and Bush (2009). Savva and Bush (2009) investigated in detail the expansion of large pores using a onedimensional lubrication model in the long wavelength limit. The lubrication model is compared against the full NS solution in Fig. 3, in which we have chosen the parameters to be identical to those used by Savva and Bush (2009) for a large pore in an inertial sheet (R0 ¼ 50, Oh¼0.04) to facilitate the comparison. Despite being a one-dimensional approximation, the lubrication model accurately captures the essential features of the film profile during the expansion of the pore, as shown by directly comparing the profiles from the full NS solutions (solid line) and from the results by Savva and Bush (2009) (circles) in Fig. 3a. As illustrated in the figure, the lubrication model is able to capture not only the formation of the large toroidal rim at the retracting
J. Lu, C.M. Corvalan / Chemical Engineering Science 132 (2015) 93–98
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Fig. 3. Dynamics of a large expanding pore. Predictions from the solution of the full Navier–Stokes system (lines) and from the lubrication model by Savva and Bush (2009) (circles) for Oh¼ 0.04 and R0 ¼ 50. (a) Film profile at inertial time t ¼ 28.2. (b) Normalized midplane velocity (solid line) and interfacial velocity (dashed line). (c) Evolution of the normalized pore tip velocity v0. Here velocities are normalized by the Taylor–Culick velocity vc.
edge of the inertial sheet but also the emergence of a series of small capillary waves ahead of the rim. The finer details of the sheet velocity, however, are not fully captured by the lubrication approximation. This discrepancy arises because the velocity profiles developed across the large toroidal rim are averaged in the lubrication approximation. Accordingly, the velocity of the sheet predicted by the lubrication model lies roughly between the interfacial velocity and the midplane velocity predicted by the solution of the NS equations (Fig. 3b). In any case, the practically important speed of the pore tip (where the midplane intersects the free interface at r ¼ r 0 ) is accurately predicted by the lubrication model. This is further illustrated for different times in Fig. 3c, which displays the excellent agreement between the full NS predictions (solid line) and the data by Savva and Bush (2009) (circles) as the speed of the pore tip increases. Note that as time increases, the tip velocity pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi approaches the limiting Taylor–Culick velocity v^ c ¼ 2σ =ρH predicted by momentum balance (Taylor, 1959; Culick, 1960). Although the results by Savva and Bush (2009) provide critical insights into the dynamics of large expanding pores, the lubrication model cannot address the dynamics of small collapsing pores due to the restriction R0 0 1 inherent in the lubrication approximation. Therefore, an important motivation for this research has been accurately describing the free-surface dynamics of small pores (R0 o1) through the solution of the full NS system.
3.2. Small collapsing pores The comparatively large radial curvature of small circular pores creates a net pull of surface tension that favors pore collapse. For the collapsing pore illustrated in Fig. 4a (and previously introduced in the summary, Fig. 2b), a dynamical evolution toward the singularity was obtained by nucleating a small pore of size R0 ¼ 0:2 in an inertial sheet with Oh¼ 0.04. The large radial curvature in the vicinity of the rim creates a region of low pressure in which the fluid is driven to the singularity at increasingly high velocity. The cross-sectional velocity field shown in Fig. 4b illustrates this dynamics at the moment when the contracting pore has reached a small radius r 0 ¼ 10 3 . A small region near the tip of the pore flows with negative radial velocity toward the center (blue in Fig. 4b), while the rest of the fluid sheet flows with positive radial velocity away from it (red in Fig. 4b). Moreover, the mid-plane sheet velocity also shown in Fig. 4b (solid line) demonstrates that only the tiny region with the largest radial curvature near the tip of the pore flows with non-negligible radial velocity, and is therefore responsible for the collapse.
3.3. Pore dynamics near the singularity Although little is known about the pore dynamics near the singularity, the large disparity between the initial and final lengthscales of the pore suggests that the contraction dynamics may depend only on the properties of the fluid sheet as r0 -0 (Barenblatt, 1996). The solution of the full NS system enabled the analysis of the evolution of the pore near collapse down to a small pore radius r 0 10 4 (or approximately 10 nm for a 100 μm thick fluid sheet). The time evolution of the pore radius r0 and the speed of collapse v0 are shown in Fig. 5 for pores with initial radius R0 ¼ 0:2 and Ohnesorge numbers sufficiently small to ensure that the dynamics is in the inertial regime. The evolution is expressed in terms of the time remaining to collapse t 0 ¼ t 0 t, where t0 is the closure time, so that t 0 -0 as r 0 -0. Once the pore size becomes sufficiently small (about one tenth of the sheet thickness), the minimum pore radius decreases following a power law r 0 ¼ ðt 0 Þα with an exponent α ¼ 0:57 for several decades (Fig. 5a). Accordingly, the collapsing speed increases as v0 ¼ αðt 0 Þα 1 (Fig. 5b). Remarkably, these power-law relationships provide not only the scaling exponents (α for radius and α 1 for velocity) but also the scaling prefactors (1 for radius and α for velocity). The predicted value of the scaling exponent α is smaller than the exponent 2=3 0:667 that results in systems in which inertia and capillarity are both important near the singularity, as occurs, for instance, during the pinch-off of inviscid drops (Leppinen and Lister, 1998, 2003; Chen and Steen, 1997). Instead, the value α ¼ 0:57 indicates a dynamics in which inertia grows asymptotically faster than capillarity, as occurs, for instance, during the pinch-off of inviscid bubbles in an ideal fluid. Indeed, for the pinch-off of axisymmetric bubbles, a time varying exponent with an effective value α ¼ 0:559 calculated from a fit to the numerical data has been predicted from asymptotic theory and simulation (Eggers et al., 2007; Leppinen et al., 2005), and exponents in the range 0:5 o α o 0:6 have been reported from experimental works (Thoroddsen et al., 2007; Keim et al., 2006). Moreover, as also demonstrated in Fig. 5, the scaling exponent is essentially unaffected by the Ohnesorge number, provided that Oh o 0:004. We have also checked that this scaling is robust to the value of the initial pore size. Results in Fig. 6 show that in the inertia limit both the scaling exponent and the amplitude are largely independent on the initial pore radius for pores with R0 r0:25. Interestingly, the pore profile becomes increasingly pointed as the pore contracts (Fig. 7a). The reason for this is that the fluid near the tip moves fastest toward the singularity because the tip has the largest radial curvature. Results from the simulation show that as the profile sharpens, the axial curvature of the tip
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Fig. 4. Dynamics of a small contracting pore. (a) Time evolution of the minimum pore radius r0 for a pore with initial radius R0 ¼ 0:2 in an inertial fluid sheet with Oh¼ 0.04. (b) Cross-sectional radial velocity with red indicating positive radial motion and blue indicating negative radial motion (top), and midplane sheet velocity for r 0 ¼ 0:001 (bottom). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
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Fig. 5. Variation of (a) minimum pore radius r0 and (b) tip velocity v0 with time to collapse t 0 for pores with initial radius R0 ¼ 0:2 in inertial sheets with Ohnesorge numbers Oh ¼ 4 10 3 (circles) and 4 10 5 (triangles). The solid lines correspond to the power-law relationships r 0 ¼ ðt 0 Þα and v0 ¼ αðt 0 Þα 1 with α ¼ 0:57.
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Fig. 6. Variation of (a) minimum pore radius r0 and (b) tip velocity v0 with time to collapse t 0 for pores with initial radii R0 ¼ 0:05 (squares), 0.15 (lower triangles), 0.20 (circles) and 0.25 (upper triangles) in an inertial sheet with Oh ¼ 4 10 3 . The solid lines correspond to the power-law relationships r 0 ¼ ðt 0 Þα and v0 ¼ αðt 0 Þα 1 with α ¼ 0:57.
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Fig. 7. Sheet profiles and axial curvature for a collapsing pore with initial radius R0 ¼ 0:2 and Oh¼ 0.004. (a) Sheet profiles for inertial times t ¼0, 0.075, 0.109, and 0.11892. The profiles shown in the inset correspond to inertial times t ¼0.11892, 0.11893 and 0.11894. (b) Evolution of the axial curvature of the tip κ nz with pore radius follows a 1=2 scaling κ nz r 0 .
κ nz κ z ðr ¼ r0 ; z ¼ 0Þ increases rapidly, following a power-law-type 1=2 behavior κ nz r 0 (Fig. 7b). The axial curvature κ z ¼ dt=ds n,
0.2
where n is the unit normal vector, and t is the unit tangent in the direction of increasing arc-length s along the interface (Slattery et al., 1990).
0
3.4. Small flickering pores The predicted profiles in Fig. 7 demonstrate that the axial curvature κz, which favors pore expansion, may increase as the pore contracts. An important aspect of this realization is that it provides a purely hydrodynamic mechanism for pore flickering. The flickering mechanism is exemplified in Fig. 8 for the pore previously introduced in the summary, Fig. 2c. Fig. 8 shows the evolution of the minimum pore radius (dashed line) and tip velocity (solid line) for a pore with initial size R0 ¼ 0:35 in an inertial sheet with Oh¼ 0.04. Given the small initial size of the pore relative to the sheet thickness (R0 o 0:5), the pore would be expected to contract and close. Remarkably, after contracting to about half of its initial radius, the pore suddenly reverses the direction of motion and expands. Fig. 8 makes plain that the pore evolves in three distinct stages. In the first stage 0 o t o0:15, the pore contracts with increasingly high speed driven by the growing radial curvature κr. As time progresses, the axial curvature κz that opposes the contraction of the pore increases. Consequently, the contraction motion slows down, and eventually stops at t 0:6. Finally, for times t 4 0:6 the direction of motion reverses (the radial velocity becomes positive), and the pore expands. For the inertial sheet in Fig. 8, the flickering mechanism is observed for initial radius in the range 0:36 r R0 r 1 (Fig. 9).
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Fig. 8. Dynamics of a small flickering pore. Evolution of the minimum pore radius r0 (dashed line) and normalized tip velocity v0 (solid line) for a pore with initial radius R0 ¼ 0:35 in an inertial sheet with Oh ¼0.04. The velocity is normalized by the Taylor–Culick velocity vc.
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4. Conclusion By solving the full Navier–Stokes system governing the behavior of small pores, we have made progress towards the understanding of the free-surface dynamics of pore contraction and collapse. We found that the size of a closing pore in an inertial sheet decreases following a simple power law scaling r 0 ¼ ðt 0 Þα with α ¼ 0:57. Remarkably, this scaling provides both the powerlaw exponent and the prefactor, so that the size of the pore and its collapsing velocity can be predicted for a given time to closure t 0 . Results further show that the critical initial size below which a
Fig. 9. Flickering pores in an inertial sheet. Evolution of the minimum pore radius r0 for pores with initial radii R0 ¼ 1 (dash-dot line), 0.5 (long-dashed line), 0.36 (solid line), and 0.35 (dashed line) in an inertial sheet with Oh¼0.04. Flickering is observed for pores with initial radius in the range 0:36 r R0 r 1.
pore will eventually collapse is unexpectedly small. The reason for this is that initially contracting pores may reverse the direction of motion (flicker) and expand. Since the origin of the power-law exponent α ¼ 0:57 cannot be explained solely by straightforward dimensional analysis, an
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important future goal is the development of asymptotic analytical solutions able to describe this apparently universal scaling near the singularity. Another important future goal is to establish the conditions under which an exterior gas will affect the closing dynamics at the incipience of collapse. As in the case of drop coalescence, pore collapse might be significantly delayed if a fluid of finite viscosity is interposed between the approaching interfaces. Finally, length scales that are absent from our study must eventually become relevant as r 0 goes to zero. For example, thermal fluctuations, which can provoke instabilities on the interface such as capillary waves, will become relevant for length scales in the order of lT ¼ kB T=σ 1=2 , where kBT is the thermal energy per unit area (Eggers, 2002). For even smaller sizes (r 0 0 lT =H), intermolecular forces will play an increasingly important role. Although better described by molecular-dynamics simulations, there have also been attempts to model intermolecular forces as body forces in the classical continuum mechanics approach (Slattery et al., 2004). Acknowledgments This project was partially supported by the Multi-University Research Initiative (MURI) (Grant no. W911NF-08-1-0171) from the U.S. Army Research Office and Purdue Research Foundation (PRF). References Barenblatt, G.I., 1996. Scaling, Self-Similarity, and Intermediate Asymptotics. Cambridge University Press, Cambridge, England. Chen, Y.J., Steen, P.H., 1997. Dynamics of inviscid capillary breakup collapse and pinchoff of a film bridge. J. Fluid Mech. 341, 245–267. Corvalan, C.M., Saita, F.A., 1991. Automatic stepsize control in continuation procedures. Comput. Chem. Eng. 15 (10), 729–739. Culick, F.E.C., 1960. Comments on ruptured soap film. J. Appl. Phys. 31 (6), 1128–1129. Debregeas, G., Gennes, G.P.D., Brochard-Wyart, F., 1998. The life and death of viscous bare bubbles. Science 279, 1704–1707. Debregeas, G., Martin, P., Brochard-Wyart, F., 1995. Viscous bursting of suspended films. Phys. Rev. Lett. 75, 3886–3889.
Dekker, C., 2007. Solid-state nanopores. Nat. Nanotechnol. 2, 209–215. Eggers, J., 2002. Dynamics of liquid nanojets. Phys. Rev. Lett. 89 (8), 084502. Eggers, J., Fontelos, M.A., Leppinen, D., Snoeijer, J.H., 2007. Theory of the collapsing axisymmetric cavity. Phys. Rev. Lett. 98, 094502. Keim, N.C., Moller, P., Zhang, W.W., Nagel, S.R., 2006. Breakup of air bubbles in water: memory and breakdown of cylindrical symmetry. Phys. Rev. Lett. 97, 144503. Keller, J.B., 1983. Breaking of liquid films and threads. Phys. Fluids 26, 3451–3453. Keller, J.B., Mikiss, M.J., 1983. Surface tension driven flows. SIAM J. Appl. Math. 43, 268–277. Kistler, S.F., Scriven, L.E., 1983. Coating Flow: Computational Analysis of Polymer Processing. Applied Science Publishers, New York. Leppinen, D., Lister, J.R., 1998. Self-similar capillary pinchoff of an inviscid fluid. Phys. Rev. Lett. 80, 704. Leppinen, D., Lister, J.R., 2003. Capillary pinch-off inviscid fluids. Phys. Fluids 15, 568. Leppinen, D., Lister, J.R., Eggers, J., 2005. Capillary pinch-off inviscid fluids at varying density ratios: the bubble limit. Bull. Am. Phys. Soc. 50 (DB.00006). Lu, J., Corvalan, C.M., 2012. Coalescence of viscous drops with surfactants. Chem. Eng. Sci. 78, 9–13. Lu, J., Corvalan, C.M., 2014. Influence of viscosity on the impingement of laminar liquid jets. Chem. Eng. Sci. 119, 182–186. Moroz, J.D., Nelson, P., 1997. Dynamically stabilized pores in bilayer membranes. Biophys. J. 72 (5), 2211–2216. Muddu, R.J., Lu, J., Sojka, P.E., Corvalan, C.M., 2012. Threshold wavelength on filaments of complex fluids. Chem. Eng. Sci. 69, 602–606. Ranz, W.E., 1950. Some experiments on the dynamics of liquid films. J. Appl. Phys. 30 (12), 1950–1955. Savva, N., Bush, J.W.M., 2009. Viscous sheet retraction. J. Fluid Mech. 626, 211–240. Schneider, G.F., Dekker, C., 2012. DNA sequencing with nanopores. Nat. Biotechnol. 30, 326–328. Slattery, J.C., 1990. Interfacial Transport Phenomena. Springer, New York. Slattery, J.C., Oh, E.-S., Fu, K., 2004. Extension of continuum mechanics to the nanoscale. Chem. Eng. Sci. 59, 4621–4635. Storm, A., Chen, J., Ling, X., Zandbergen, H., Dekker, C., 2005. Electron-beaminduced deformations of SiO2 nanostructures. J. Appl. Phys. 98 (1), 014307. Storm, A.J., Chen, J.H., Ling, X.S., Zandbergen, H.W., Dekker, C., 2003. Fabrication of solid-state nanopores with single-nanometre precision. Nat. Mater. 2, 537–540. Taylor, G.I., 1959. The dynamics of thin sheets of fluid. III. Disintegration of fluid sheets. Proc. R. Soc. Lond. A 253, 313–321. Taylor, G.I., Michael, D.H., 1973. On making holes in a sheet of fluid. J. Fluid Mech. 4 (58), 625. Thoroddsen, S.T., Etoh, T.G., Takehara, K., 2007. Experiments on bubble pinch-off. Phys. Fluids 19, 042101. Wanunu, M., 2012. Nanopores: a journey towards DNA sequencing. Phys. Life Rev. 9 (2), 125–158. Xue, Z., Corvalan, C.M., Dravid, V., Sojka, P.E., 2008. Breakup of shear-thinning liquid jets with surfactants. Chem. Eng. Sci. 63 (7), 1842–1849. Zhao, X.P., Zhang, M.J., Yang, R.T., 2010. Control of pore radius regulation for electroporation-based drug delivery. Commun. Nonlinear Sci. Numer. Simul. 15, 1400–1407.
Contents lists available at ScienceDirect
Chemical Engineering Science journal homepage: www.elsevier.com/locate/ces
Free-surface dynamics of small pores Jiakai Lu, Carlos M. Corvalan n Transport Phenomena Laboratory, Department of Food Science, Purdue University, West Lafayette, IN 47907, USA
H I G H L I G H T S
We consider the contraction of small pores nucleated in fluid sheets. We characterize the pore dynamics solving the full Navier–Stokes system. A scaling function without free parameters describes the size of closing pores near collapse. Contracting pores may reverse the direction of motion (flicker) and expand.
art ic l e i nf o
a b s t r a c t
Article history: Received 17 January 2015 Received in revised form 28 March 2015 Accepted 4 April 2015 Available online 17 April 2015
When the size of a pore nucleated in a fluid sheet is sufficiently small, the pore will contract and close driven by its large radial curvature. The dynamics of small contracting pores are relevant to a number of natural systems and practical applications, from fission pores in cell membranes to the fabrication of nanopores in sensors for DNA sequencing. Here, we report high-fidelity numerical simulations that provide detailed insight into the mechanisms of pore contraction and collapse in fluid sheets of low viscosity. Results uncover a scaling law that predicts the radius of a closing pore as a function of the time to collapse without free parameters. Simulations also show that contracting pores do not always proceed to collapse. Instead, some contracting pores reverse the direction of motion and expand. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Nanopore Pore Interface Simulation Transport processes Cell membrane
1. Introduction This paper reports a numerical study of the free-surface dynamics of small pores in fluid sheets. Unlike large pores, sufficiently small pores contract and close driven by surface tension and their large radial curvature—a behavior discussed in the pioneering work by Taylor and Michael (1973). The dynamics of small contracting pores in fluid sheets impact a wide range of systems, from electroporation and fission pores in cell membranes to the stability of food foams (Moroz and Nelson, 1997; Wanunu, 2012; Zhao et al., 2010). Moreover, understanding and ultimately controlling the dynamics of small pores in the micro and nanoscales offer potential applications for the fabrication of sensors for rapid characterization of biomolecules and DNA sequencing (Storm et al., 2003, 2005; Dekker, 2007; Schneider and Dekker, 2012).
n
Corresponding author. E-mail address: [email protected] (C.M. Corvalan).
http://dx.doi.org/10.1016/j.ces.2015.04.013 0009-2509/& 2015 Elsevier Ltd. All rights reserved.
Although the dynamics of large expanding pores have been extensively studied and the speed of expansion is now well characterized (Culick, 1960; Taylor, 1959; Ranz, 1950; Savva and Bush, 2009; Debregeas et al., 1995, 1998; Keller et al., 1983; Keller and Mikiss, 1983), surprisingly little is known about the freesurface dynamics of small contracting pores and their speed of collapse. Indeed, in a recent theoretical and numerical work, Savva and Bush (2009) investigated the free-surface dynamics of circular pores using a lubrication model in the long wavelength limit. Their findings provided important insights into the early stage dynamics of pore expansion, and confirmed the exponential growth rate observed in experiments. However, due to limitations intrinsic to the lubrication approximation, the lubrication model cannot predict the dynamics of small contracting pores. To study the free-surface dynamics of small pores, we have developed high-fidelity simulations that overcome the limitations of the lubrication model by solving the full Navier–Stokes (NS) equations. Here, we report how the solution of the full NS system enables a detailed analysis of the free-surface dynamics of pore contraction (Section 3.2), and uncovers a new scaling law that predicts the size of collapsing pores near the singularity without
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free parameters (Section 3.3). Results also characterize the behavior of small contracting pores that suddenly change the direction of motion and expand (flicker) by purely hydrodynamic reasons (Section 3.4).
2. Problem description We now state our model and assumptions. We consider the free-surface dynamics of circular pores nucleated in a fluid sheet of density ρ, viscosity μ, and surface tension σ, as sketched in Fig. 1. The model and results are described in this paper using the thickness of the fluid sheet H as characteristic length scale, the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi inviscid time τ ρH 3 =σ as characteristic time scale, and the
stress μ=τ as characteristic pressure scale. To analyze the pore dynamics, we numerically solved the full axisymmetric Navier–Stokes and continuity equations: ∂v þ v ∇v ¼ ∇p þ ∇2 v; ð1Þ Re ∂t ∇ v ¼ 0;
ð2Þ
for the velocity v and pressure p. The pReynolds number ffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi Re ρσ H =μ based on the inviscid pffiffiffi velocity σ =ρH is related to the Ohnesorgepnumber ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Oh ¼ 1= 2 Re based on the Taylor–Culick velocity v^ c ¼ 2σ =ρH (Taylor, 1959; Culick, 1960). We further assume that gravity can be neglected compared with surface tension. Along the sheet interface both the traction boundary condition T n ¼ 2 Re Hn;
0.3 0.2 0.1
ð3Þ
0
and the kinematic boundary condition n ðv vs Þ ¼ 0
Fig. 1. Small circular pore on a fluid sheet with density ρ, viscosity μ, and surface tension σ. The thickness of the fluid sheet is H and the initial pore radius is R0.
ð4Þ
were imposed, where T is the stress tensor, vs the velocity of the points at the interface, and n the unit normal vector to the interface (Slattery et al., 1990). The interfacial curvature is H ¼ 1=2 ðκ z þ κ r Þ, where κz and κr are the axial and radial curvature, respectively (Slattery et al., 1990). In addition, symmetry boundary conditions were imposed on the plane z¼ 0 and axis of symmetry r ¼ 0. We used the finite-element method along with the arbitrary Lagrangian–Eulerian method of spines pioneered by Kistler and Scriven (1983) to parametrize the spatial derivatives and the deforming pore interface, following a design that we have successfully applied to similar free-surface flows (Xue et al., 2008; Muddu et al., 2012; Lu and Corvalan, 2012, 2014). The time was discretized using a second-order trapezoidal method with Adam–Bashforth prediction to reduce time truncation errors. To improve computational efficiency, the time steps were calculated using a first-order continuation method (Corvalan and Saita, 1991).
3. Results and discussion The qualitative behavior of a pore nucleated in a fluid sheet may change substantially with the initial pore size. There are at least four conceivable behaviors: the pore may open, close, flicker or remain stable. Here we discuss three of these scenarios, which are summarized in Fig. 2. The figure compares the time evolution of three different pores which open (long dashed line), close (dashed line), or flicker (solid line) depending solely upon the initial pore size R0. We have not observed stable pores (in the sense of Taylor and Michael, 1973) for the parameters considered in this paper.
-0.1 -0.2 -0.3
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Fig. 2. Influence of initial size on the dynamics of pore formation. Time evolution of the minimum pore radius r0 in an inertial fluid sheet with Oh¼ 0.04. Depending on the initial radius R0, the pore: (a) expands (long-dashed line, R0 ¼ 50), (b) contracts (dashed line, R0 ¼ 0:2), or (c) initially contracts and then expands (solid line, R0 ¼ 0:36).
3.1. A large expanding pore Although our focus is on the little-known dynamics of small pores, we begin our discussion with the large opening pore introduced in Fig. 2a in order to make comparisons with a recent work by Savva and Bush (2009). Savva and Bush (2009) investigated in detail the expansion of large pores using a onedimensional lubrication model in the long wavelength limit. The lubrication model is compared against the full NS solution in Fig. 3, in which we have chosen the parameters to be identical to those used by Savva and Bush (2009) for a large pore in an inertial sheet (R0 ¼ 50, Oh¼0.04) to facilitate the comparison. Despite being a one-dimensional approximation, the lubrication model accurately captures the essential features of the film profile during the expansion of the pore, as shown by directly comparing the profiles from the full NS solutions (solid line) and from the results by Savva and Bush (2009) (circles) in Fig. 3a. As illustrated in the figure, the lubrication model is able to capture not only the formation of the large toroidal rim at the retracting
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Fig. 3. Dynamics of a large expanding pore. Predictions from the solution of the full Navier–Stokes system (lines) and from the lubrication model by Savva and Bush (2009) (circles) for Oh¼ 0.04 and R0 ¼ 50. (a) Film profile at inertial time t ¼ 28.2. (b) Normalized midplane velocity (solid line) and interfacial velocity (dashed line). (c) Evolution of the normalized pore tip velocity v0. Here velocities are normalized by the Taylor–Culick velocity vc.
edge of the inertial sheet but also the emergence of a series of small capillary waves ahead of the rim. The finer details of the sheet velocity, however, are not fully captured by the lubrication approximation. This discrepancy arises because the velocity profiles developed across the large toroidal rim are averaged in the lubrication approximation. Accordingly, the velocity of the sheet predicted by the lubrication model lies roughly between the interfacial velocity and the midplane velocity predicted by the solution of the NS equations (Fig. 3b). In any case, the practically important speed of the pore tip (where the midplane intersects the free interface at r ¼ r 0 ) is accurately predicted by the lubrication model. This is further illustrated for different times in Fig. 3c, which displays the excellent agreement between the full NS predictions (solid line) and the data by Savva and Bush (2009) (circles) as the speed of the pore tip increases. Note that as time increases, the tip velocity pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi approaches the limiting Taylor–Culick velocity v^ c ¼ 2σ =ρH predicted by momentum balance (Taylor, 1959; Culick, 1960). Although the results by Savva and Bush (2009) provide critical insights into the dynamics of large expanding pores, the lubrication model cannot address the dynamics of small collapsing pores due to the restriction R0 0 1 inherent in the lubrication approximation. Therefore, an important motivation for this research has been accurately describing the free-surface dynamics of small pores (R0 o1) through the solution of the full NS system.
3.2. Small collapsing pores The comparatively large radial curvature of small circular pores creates a net pull of surface tension that favors pore collapse. For the collapsing pore illustrated in Fig. 4a (and previously introduced in the summary, Fig. 2b), a dynamical evolution toward the singularity was obtained by nucleating a small pore of size R0 ¼ 0:2 in an inertial sheet with Oh¼ 0.04. The large radial curvature in the vicinity of the rim creates a region of low pressure in which the fluid is driven to the singularity at increasingly high velocity. The cross-sectional velocity field shown in Fig. 4b illustrates this dynamics at the moment when the contracting pore has reached a small radius r 0 ¼ 10 3 . A small region near the tip of the pore flows with negative radial velocity toward the center (blue in Fig. 4b), while the rest of the fluid sheet flows with positive radial velocity away from it (red in Fig. 4b). Moreover, the mid-plane sheet velocity also shown in Fig. 4b (solid line) demonstrates that only the tiny region with the largest radial curvature near the tip of the pore flows with non-negligible radial velocity, and is therefore responsible for the collapse.
3.3. Pore dynamics near the singularity Although little is known about the pore dynamics near the singularity, the large disparity between the initial and final lengthscales of the pore suggests that the contraction dynamics may depend only on the properties of the fluid sheet as r0 -0 (Barenblatt, 1996). The solution of the full NS system enabled the analysis of the evolution of the pore near collapse down to a small pore radius r 0 10 4 (or approximately 10 nm for a 100 μm thick fluid sheet). The time evolution of the pore radius r0 and the speed of collapse v0 are shown in Fig. 5 for pores with initial radius R0 ¼ 0:2 and Ohnesorge numbers sufficiently small to ensure that the dynamics is in the inertial regime. The evolution is expressed in terms of the time remaining to collapse t 0 ¼ t 0 t, where t0 is the closure time, so that t 0 -0 as r 0 -0. Once the pore size becomes sufficiently small (about one tenth of the sheet thickness), the minimum pore radius decreases following a power law r 0 ¼ ðt 0 Þα with an exponent α ¼ 0:57 for several decades (Fig. 5a). Accordingly, the collapsing speed increases as v0 ¼ αðt 0 Þα 1 (Fig. 5b). Remarkably, these power-law relationships provide not only the scaling exponents (α for radius and α 1 for velocity) but also the scaling prefactors (1 for radius and α for velocity). The predicted value of the scaling exponent α is smaller than the exponent 2=3 0:667 that results in systems in which inertia and capillarity are both important near the singularity, as occurs, for instance, during the pinch-off of inviscid drops (Leppinen and Lister, 1998, 2003; Chen and Steen, 1997). Instead, the value α ¼ 0:57 indicates a dynamics in which inertia grows asymptotically faster than capillarity, as occurs, for instance, during the pinch-off of inviscid bubbles in an ideal fluid. Indeed, for the pinch-off of axisymmetric bubbles, a time varying exponent with an effective value α ¼ 0:559 calculated from a fit to the numerical data has been predicted from asymptotic theory and simulation (Eggers et al., 2007; Leppinen et al., 2005), and exponents in the range 0:5 o α o 0:6 have been reported from experimental works (Thoroddsen et al., 2007; Keim et al., 2006). Moreover, as also demonstrated in Fig. 5, the scaling exponent is essentially unaffected by the Ohnesorge number, provided that Oh o 0:004. We have also checked that this scaling is robust to the value of the initial pore size. Results in Fig. 6 show that in the inertia limit both the scaling exponent and the amplitude are largely independent on the initial pore radius for pores with R0 r0:25. Interestingly, the pore profile becomes increasingly pointed as the pore contracts (Fig. 7a). The reason for this is that the fluid near the tip moves fastest toward the singularity because the tip has the largest radial curvature. Results from the simulation show that as the profile sharpens, the axial curvature of the tip
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0.3
0.2 0 -0.5
0.1
-1 0
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1
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2
Fig. 4. Dynamics of a small contracting pore. (a) Time evolution of the minimum pore radius r0 for a pore with initial radius R0 ¼ 0:2 in an inertial fluid sheet with Oh¼ 0.04. (b) Cross-sectional radial velocity with red indicating positive radial motion and blue indicating negative radial motion (top), and midplane sheet velocity for r 0 ¼ 0:001 (bottom). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
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Fig. 5. Variation of (a) minimum pore radius r0 and (b) tip velocity v0 with time to collapse t 0 for pores with initial radius R0 ¼ 0:2 in inertial sheets with Ohnesorge numbers Oh ¼ 4 10 3 (circles) and 4 10 5 (triangles). The solid lines correspond to the power-law relationships r 0 ¼ ðt 0 Þα and v0 ¼ αðt 0 Þα 1 with α ¼ 0:57.
0
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-1
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-2
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1
-3
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Fig. 6. Variation of (a) minimum pore radius r0 and (b) tip velocity v0 with time to collapse t 0 for pores with initial radii R0 ¼ 0:05 (squares), 0.15 (lower triangles), 0.20 (circles) and 0.25 (upper triangles) in an inertial sheet with Oh ¼ 4 10 3 . The solid lines correspond to the power-law relationships r 0 ¼ ðt 0 Þα and v0 ¼ αðt 0 Þα 1 with α ¼ 0:57.
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0
1.5 1
150
0.02
*
-0.02 -0.005
0
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50
0 -0.5 -0.6
-0.3
0
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0.6
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0.2
Fig. 7. Sheet profiles and axial curvature for a collapsing pore with initial radius R0 ¼ 0:2 and Oh¼ 0.004. (a) Sheet profiles for inertial times t ¼0, 0.075, 0.109, and 0.11892. The profiles shown in the inset correspond to inertial times t ¼0.11892, 0.11893 and 0.11894. (b) Evolution of the axial curvature of the tip κ nz with pore radius follows a 1=2 scaling κ nz r 0 .
κ nz κ z ðr ¼ r0 ; z ¼ 0Þ increases rapidly, following a power-law-type 1=2 behavior κ nz r 0 (Fig. 7b). The axial curvature κ z ¼ dt=ds n,
0.2
where n is the unit normal vector, and t is the unit tangent in the direction of increasing arc-length s along the interface (Slattery et al., 1990).
0
3.4. Small flickering pores The predicted profiles in Fig. 7 demonstrate that the axial curvature κz, which favors pore expansion, may increase as the pore contracts. An important aspect of this realization is that it provides a purely hydrodynamic mechanism for pore flickering. The flickering mechanism is exemplified in Fig. 8 for the pore previously introduced in the summary, Fig. 2c. Fig. 8 shows the evolution of the minimum pore radius (dashed line) and tip velocity (solid line) for a pore with initial size R0 ¼ 0:35 in an inertial sheet with Oh¼ 0.04. Given the small initial size of the pore relative to the sheet thickness (R0 o 0:5), the pore would be expected to contract and close. Remarkably, after contracting to about half of its initial radius, the pore suddenly reverses the direction of motion and expands. Fig. 8 makes plain that the pore evolves in three distinct stages. In the first stage 0 o t o0:15, the pore contracts with increasingly high speed driven by the growing radial curvature κr. As time progresses, the axial curvature κz that opposes the contraction of the pore increases. Consequently, the contraction motion slows down, and eventually stops at t 0:6. Finally, for times t 4 0:6 the direction of motion reverses (the radial velocity becomes positive), and the pore expands. For the inertial sheet in Fig. 8, the flickering mechanism is observed for initial radius in the range 0:36 r R0 r 1 (Fig. 9).
-0.2 -0.4 -0.6
0
0.2
0.4
0.6
0.8
1
Fig. 8. Dynamics of a small flickering pore. Evolution of the minimum pore radius r0 (dashed line) and normalized tip velocity v0 (solid line) for a pore with initial radius R0 ¼ 0:35 in an inertial sheet with Oh ¼0.04. The velocity is normalized by the Taylor–Culick velocity vc.
0.2
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4. Conclusion By solving the full Navier–Stokes system governing the behavior of small pores, we have made progress towards the understanding of the free-surface dynamics of pore contraction and collapse. We found that the size of a closing pore in an inertial sheet decreases following a simple power law scaling r 0 ¼ ðt 0 Þα with α ¼ 0:57. Remarkably, this scaling provides both the powerlaw exponent and the prefactor, so that the size of the pore and its collapsing velocity can be predicted for a given time to closure t 0 . Results further show that the critical initial size below which a
Fig. 9. Flickering pores in an inertial sheet. Evolution of the minimum pore radius r0 for pores with initial radii R0 ¼ 1 (dash-dot line), 0.5 (long-dashed line), 0.36 (solid line), and 0.35 (dashed line) in an inertial sheet with Oh¼0.04. Flickering is observed for pores with initial radius in the range 0:36 r R0 r 1.
pore will eventually collapse is unexpectedly small. The reason for this is that initially contracting pores may reverse the direction of motion (flicker) and expand. Since the origin of the power-law exponent α ¼ 0:57 cannot be explained solely by straightforward dimensional analysis, an
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important future goal is the development of asymptotic analytical solutions able to describe this apparently universal scaling near the singularity. Another important future goal is to establish the conditions under which an exterior gas will affect the closing dynamics at the incipience of collapse. As in the case of drop coalescence, pore collapse might be significantly delayed if a fluid of finite viscosity is interposed between the approaching interfaces. Finally, length scales that are absent from our study must eventually become relevant as r 0 goes to zero. For example, thermal fluctuations, which can provoke instabilities on the interface such as capillary waves, will become relevant for length scales in the order of lT ¼ kB T=σ 1=2 , where kBT is the thermal energy per unit area (Eggers, 2002). For even smaller sizes (r 0 0 lT =H), intermolecular forces will play an increasingly important role. Although better described by molecular-dynamics simulations, there have also been attempts to model intermolecular forces as body forces in the classical continuum mechanics approach (Slattery et al., 2004). Acknowledgments This project was partially supported by the Multi-University Research Initiative (MURI) (Grant no. W911NF-08-1-0171) from the U.S. Army Research Office and Purdue Research Foundation (PRF). References Barenblatt, G.I., 1996. Scaling, Self-Similarity, and Intermediate Asymptotics. Cambridge University Press, Cambridge, England. Chen, Y.J., Steen, P.H., 1997. Dynamics of inviscid capillary breakup collapse and pinchoff of a film bridge. J. Fluid Mech. 341, 245–267. Corvalan, C.M., Saita, F.A., 1991. Automatic stepsize control in continuation procedures. Comput. Chem. Eng. 15 (10), 729–739. Culick, F.E.C., 1960. Comments on ruptured soap film. J. Appl. Phys. 31 (6), 1128–1129. Debregeas, G., Gennes, G.P.D., Brochard-Wyart, F., 1998. The life and death of viscous bare bubbles. Science 279, 1704–1707. Debregeas, G., Martin, P., Brochard-Wyart, F., 1995. Viscous bursting of suspended films. Phys. Rev. Lett. 75, 3886–3889.
Dekker, C., 2007. Solid-state nanopores. Nat. Nanotechnol. 2, 209–215. Eggers, J., 2002. Dynamics of liquid nanojets. Phys. Rev. Lett. 89 (8), 084502. Eggers, J., Fontelos, M.A., Leppinen, D., Snoeijer, J.H., 2007. Theory of the collapsing axisymmetric cavity. Phys. Rev. Lett. 98, 094502. Keim, N.C., Moller, P., Zhang, W.W., Nagel, S.R., 2006. Breakup of air bubbles in water: memory and breakdown of cylindrical symmetry. Phys. Rev. Lett. 97, 144503. Keller, J.B., 1983. Breaking of liquid films and threads. Phys. Fluids 26, 3451–3453. Keller, J.B., Mikiss, M.J., 1983. Surface tension driven flows. SIAM J. Appl. Math. 43, 268–277. Kistler, S.F., Scriven, L.E., 1983. Coating Flow: Computational Analysis of Polymer Processing. Applied Science Publishers, New York. Leppinen, D., Lister, J.R., 1998. Self-similar capillary pinchoff of an inviscid fluid. Phys. Rev. Lett. 80, 704. Leppinen, D., Lister, J.R., 2003. Capillary pinch-off inviscid fluids. Phys. Fluids 15, 568. Leppinen, D., Lister, J.R., Eggers, J., 2005. Capillary pinch-off inviscid fluids at varying density ratios: the bubble limit. Bull. Am. Phys. Soc. 50 (DB.00006). Lu, J., Corvalan, C.M., 2012. Coalescence of viscous drops with surfactants. Chem. Eng. Sci. 78, 9–13. Lu, J., Corvalan, C.M., 2014. Influence of viscosity on the impingement of laminar liquid jets. Chem. Eng. Sci. 119, 182–186. Moroz, J.D., Nelson, P., 1997. Dynamically stabilized pores in bilayer membranes. Biophys. J. 72 (5), 2211–2216. Muddu, R.J., Lu, J., Sojka, P.E., Corvalan, C.M., 2012. Threshold wavelength on filaments of complex fluids. Chem. Eng. Sci. 69, 602–606. Ranz, W.E., 1950. Some experiments on the dynamics of liquid films. J. Appl. Phys. 30 (12), 1950–1955. Savva, N., Bush, J.W.M., 2009. Viscous sheet retraction. J. Fluid Mech. 626, 211–240. Schneider, G.F., Dekker, C., 2012. DNA sequencing with nanopores. Nat. Biotechnol. 30, 326–328. Slattery, J.C., 1990. Interfacial Transport Phenomena. Springer, New York. Slattery, J.C., Oh, E.-S., Fu, K., 2004. Extension of continuum mechanics to the nanoscale. Chem. Eng. Sci. 59, 4621–4635. Storm, A., Chen, J., Ling, X., Zandbergen, H., Dekker, C., 2005. Electron-beaminduced deformations of SiO2 nanostructures. J. Appl. Phys. 98 (1), 014307. Storm, A.J., Chen, J.H., Ling, X.S., Zandbergen, H.W., Dekker, C., 2003. Fabrication of solid-state nanopores with single-nanometre precision. Nat. Mater. 2, 537–540. Taylor, G.I., 1959. The dynamics of thin sheets of fluid. III. Disintegration of fluid sheets. Proc. R. Soc. Lond. A 253, 313–321. Taylor, G.I., Michael, D.H., 1973. On making holes in a sheet of fluid. J. Fluid Mech. 4 (58), 625. Thoroddsen, S.T., Etoh, T.G., Takehara, K., 2007. Experiments on bubble pinch-off. Phys. Fluids 19, 042101. Wanunu, M., 2012. Nanopores: a journey towards DNA sequencing. Phys. Life Rev. 9 (2), 125–158. Xue, Z., Corvalan, C.M., Dravid, V., Sojka, P.E., 2008. Breakup of shear-thinning liquid jets with surfactants. Chem. Eng. Sci. 63 (7), 1842–1849. Zhao, X.P., Zhang, M.J., Yang, R.T., 2010. Control of pore radius regulation for electroporation-based drug delivery. Commun. Nonlinear Sci. Numer. Simul. 15, 1400–1407.