Molecular Dynamics Simulations of Capillary Rise Experiments in

Bosanquet, C. H. Philos. Mag. 1923, 45, 525. [Crossref], [CAS]. (23) . ...... John Ralston , Mihail Popescu , Rossen Sedev. Annual Review of Materials...
1 downloads 0 Views 708KB Size
Download PDF
1232

Langmuir 2008, 24, 1232-1239

Molecular Dynamics Simulations of Capillary Rise Experiments in Nanotubes Coated with Polymer Brushes† D. I. Dimitrov,‡ A. Milchev,‡,§ and K. Binder*,‡ Institut fu¨r Physik, Johannes Gutenberg UniVersita¨t Mainz, Staudinger, Weg 7, 55099 Mainz, Germany, and Instute for Physical Chemistry, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria ReceiVed June 29, 2007. In Final Form: August 9, 2007 The capillary filling of a nanotube coated with a polymer brush is studied by molecular dynamics simulations of a coarse-grained model, assuming various conditions for the fluid-wall and fluid-brush interactions. Whereas the fluid is modeled by simple point particles interacting with Lennard-Jones forces, the (end-grafted, fully flexible) polymers that form the brush coating are described by a standard bead-spring model. Our experiments reveal that capillary filling is observed even for walls that would not be wetted by the fluid, provided the polymer brush coating itself wets. Generally, it is found that the capillary rise always proceeds through a t1/2 law with time t while the underlying molecular mechanism differs for wettable and nonwettable walls. For wettable walls, fluid imbibition is compatible with the Lucas-Washburn mechanism whereby the total influx of matter drops steadily with growing chain length N and the meniscus speed goes through a minimum at intermediate chain lengths. Moreover, because of flow, the polymer brush reorganizes its structure by forming a dense plug of chain segments under the meniscus that follows the meniscus in its motion. When the tube wall does not wet, one observes no meniscus formation for short chains although the fluid seeps through the wet brush. For a brush coating with longer chains, axial segregation between the brush segments and the fluid occurs by a kind of diffusive spreading, reminiscent of invasion percolation transport in a random medium, leading to the formation of a moving meniscus. For even longer chains that reach the tube axis, the rise of a meniscus with vanishing curvature-like imbibition in a porous medium is observed to take place.

1. Introduction nanoscale1-4

Fluid flow on the is a fascinating phenomenon with many important applications in nanotechnology. Nanofluidic devices play a role in developments such as the lab-on-a-chip whereas fluid flow through pores in biomembranes is a central problem in living matter and flow through the irregular pore network of porous materials is a key process in separation technologies and oil recovery. A key issue is to understand how one can control the flow properties of the fluid in the nanopore by a suitable modification of the surface forces acting from the pore walls on the liquid particles.5 A well-known recipe for modifying surface-liquid interactions consists of coating the surface with end-grafted flexible polymers, creating the so-called polymer brushes.6-11 Thus, polymer brushes on the surface of colloidal particles stabilize colloidal suspensions,12 and on flat surfaces, they act as very efficient lubricants, reducing frictional forces.13-17 End-grafted polyelectrolytes have been studied as an effective means of controlling electroosmotic flow.18,19 †

Part of the Molecular and Surface Forces special issue. Johannes Gutenberg Universita¨t Mainz. § Bulgarian Academy of Sciences. ‡

(1) Meller, A. J. Phys.: Condens. Matter 2003, 15, R581. (2) Squires, T. M.; Quake, S. R. ReV. Mod. Phys. 2005, 77, 977. (3) Rouquerol, F.; Rouquerol, J.; Sing, K. Adsorption by Powders and Porous Solids; Academic Press: New York, 1999. (4) Huber, P.; Gru¨ner, S.; Scha¨fer, C.; Knorr, K. and Kityk, A. V. Eur. Phys. J. Spec. Top. 2007, 141, 101. (5) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: London, 1991. (6) Polymer Brushes; Advincula, R. C., Brittain, W. J., Caster, K. C., Ru¨he, J., Eds.; Wiley-VCH: Weinheim, Germany, 2004. (7) Halperin, A.; Tirell, M.; Lodge, T. P. AdV. Polym. Sci. 1991, 100, 31. (8) Milner, S. T. Science 1991, 251, 905. (9) Szleifer, I.; Carignano, M. A. AdV. Chem. Phys. 1996, 94, 165. (10) Grest, G. S. AdV. Polym. Sci. 1999, 138, 149. (11) Le´ger, L.; Raphael, E.; Hervet, H. AdV. Polym. Sci. 1999, 138, 185. (12) Napper D. H. Polymeric Stabilization of Colloidal Dispersions; Academic Press: London, 1983. (13) Klein, J. In Liquids at Interfaces; Charvolin, J., Joanny, J. F., Zinn-Justin, J., Eds.; Elsevier Science: Amsterdam, 1990.

In this article, we study the effect of modifying the surface forces of a nanoscopically narrow capillary wall, covered by end-grafted polymers, on the kinetics of capillary filling. Since the pioneering work of Lucas20 and Washburn,21 numerous attempts have been made to understand the rise of liquids in capillary tubes theoretically (with some experimental studies being carried out more than a century ago22). On the macroscopic scale, a basic understanding of the capillary rise processes is provided by an equation now referred to as the Lucas-Washburn equation,20-28 which has been derived for the case of Poiseille flow in a cylindrical capillary:

H(t) )

(

)

γLVR cos θ 2η

1/2

xt

(1)

Equation 1 assumes the existence of a sharp interface between the liquid and its vapor, and in its derivation, transient effects (14) Klein, J.; Kumacheva, E.; Perahia, D.; Mahalu, D.; Warburg, S. Faraday Discuss. 1994, 98, 173. (15) Granick, S.; Demirel, A. L.; Cai, L.; and Peanasky, J. Isr. J. Chem. 1995, 35, 75. (16) Klein, J. Ann. ReV. Mater. Sci. 1996, 26, 581. (17) Kreer, T.; Muser, M. H.; Binder, K.; Klein, J. Langmuir 2001, 17, 7804. (18) Tessier, F.; Slater, G. W. Macromolecules 2005, 38, 6752. Tessier, F.; Slater, G. W. Macromolecules 2006, 39, 1250. (19) Harden, J. L.; Long, D.; Ajdari, A. Langmuir 2001, 17, 705. (20) Lucas, R. Kolloid Z. 1918, 23, 15. (21) Washburn, E. W. Phys. ReV. 1921, 17, 273. (22) Bell, J. M.; Cameron, F. K. J. Phys. Chem. 1996, 10, 658. (23) Bosanquet, C. H. Philos. Mag. 1923, 45, 525. (24) Siegel, R. J. Appl. Mech. 1961, 83, 165. (25) Marmur, A. In Modern Approach to Wettability: Theory and Applications; Schrader, M. E., Loeb, G, Eds.; Plenum Publishing: New York, 1992; pp 327358. (26) Zhmud, B. V.; Tiberg, F.; and Hallstensson, K. J. Colloid Interface Sci. 2000, 228, 263. (27) Martic, G.; Gentner, F.; Seveno, D.; Coulon, D.; De Coninck, J.; Blake, T. D. Langmuir 2002, 18, 7971. (28) Kornev, K. G.; Neimark, A. V. J. Colloid Interface Sci. 2003, 262, 253.

10.1021/la7019445 CCC: $40.75 © 2008 American Chemical Society Published on Web 10/05/2007

MD Simulations of Capillary Rise Experiments

Figure 1. Typical snapshot showing a cross section of the capillary in the beginning of the imbibition process. The solvent particles are semitransparent, so the polymer brush is also visible. Chains of length N ) 16 coat the inner wall of the tube at grafting density σG ) 0.05. The fluid is kept in a closed container at the tube opening by an attractive wall.

Langmuir, Vol. 24, No. 4, 2008 1233

complete wetting,29,30 θ ) 0, whereas for incomplete wetting (of a lyophilic wall), 0 < θ < 90°. Of course, eq 1 cannot be true for t f 0 (where a final force applied to an infinitesimal mass would lead to a singularity in liquid front acceleration) but holds only after a (nanoscopically small32) transient time. Zhmud et al.26 and Huang et al.31 suggest an initial behavior of H(t) ∝ t2 when the liquid is accelerated by the capillary forces. However, Martic et al.,50 performing molecular dynamics (MD)33-35 simulations of a polymer melt, did not confirm this rise of the meniscus height with the square of time but rather found H(t) rising more slowly than linearly with time, even for t ≈ 1 ns. Martic et al.27 suggested a slight correction to eq 1 by replacing θ with a dynamic contact angle.29,36,37 The concept of a dynamic contact angle also proved to be valuable for the interpretation of large-scale MD simulations of imbibition into a pore with a heterogeneous surface.38 Other suggestions (believed to be relevant to viscoelastic fluids such as blood flowing in blood vessels) were made by Kornev and Neimark, accounting for the shear flow deformation caused by the velocity gradients (Weissenberg effect39). Finally, on the basis of observations for a simple fluid in a carbon nanotube, Supple and Quirke40,41,4 expressed doubts about the validity of eq 1 on the nanoscale because their results implied a simple linear law H(t) ∝ t. However, the latter result is in agreement neither with recent experiments4,42 nor with extensive simulations of an LJ fluid and the coarse-grained model of a polymer melt43 where the wall-liquid attraction strength has been varied over a wide range and the H(t) ∝ xt relation was obtained. In these simulations,43 we have also determined the parameters γLV and η that enter the prefactor in eq 1 and have thus demonstrated that (in the case of an LJ fluid) eq 1 is indeed a qualitatively reliable description for the studied model even on the nanoscale. For the case of a polymer melt, however, slip flow occurs at the walls as expected,44,45 and for this case, an extension of the Lucas-Washburn equation was proposed43

H(t) )

[

]

γLV(R + δ)2 cos θ 2Rη

1/2

xt

(2)

Here δ is the slip length45 characterizing the hydrodynamic boundary condition of the flowing fluid at the capillary wall.

Figure 2. Squared meniscus height vs time for a brush-coated wettable nanotube. The chains have lengths of N ) 8 (a) and 24 (b). The brush degree of wettability pl is given in part (a) as a parameter. In the inset, one can see the change in the slope (i.e., the “speed” of variation of the squared meniscus height, dH2/dt) with the growing wettability of the coating. The real speed of the meniscus is dH/dt, and it decreases with time.

such as the initial burst of fluid in the capillary are completely neglected. Assuming further a steady laminar flow, eq 1 predicts for the rise of height H(t) of the fluid meniscus with time t a xt law also yielding the prefactor of this law: γLV is the surface tension of the liquid, η is its shear viscosity, R is the radius of the pore, and θ is the contact angle between the meniscus and the wall. It is the latter quantity that is controlled by the surface forces between the wall and the fluid atoms, of course. For

(29) De Gennes, P. G. ReV. Mod. Phys. 1985, 57, 827. (30) Dietrich, S. In Phase Transitions and Critical Phenomena; Domb C., Lebowitz, J. L., Eds.; Academic Press: New York, 1988; Vol. 12, pp 1-218. (31) Huang, W.; Liu, Q.; Li, Y. Chem. Eng. Technol. 2006, 29, 716. (32) De Gennes, G. Capillarity and Wetting Phenomena; Springer: Berlin, 2004. (33) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon Press: Oxford, 1987. (34) Rapaport, D. C. The Art of Molecular Dynamics Simulation; Cambridge University Press: Cambridge, 1987. (35) Sprik, M. In Monte Carlo and Molecular Dynamics of Condensed Matter Systems; Binder, K., Ciccotti, G., Eds.; Editrice Compositori: Bologna, Italy, 1996; pp 43-88. (36) Gerdes, S.; Cazabat, A.-M.; Strom, G.; Tiberg, F. Langmuir 1998, 14, 7052. (37) Blake, T. D.; De Coninck, J. J. AdV. Colloid Interface Sci. 2002, 96, 21. (38) Martic, G.; Blake, T. D.; De Coninck, J. Langmuir 2005, 21, 11201. (39) Weissenberg, K. Nature 1947, 159, 310. (40) Supple, S.; Quirke, N. Phys. ReV. Lett. 2003, 90, 214501. (41) Supple, S.; Quirke, N. J. Chem. Phys. 2004, 121, 8571. (42) Huber, P. J. Mater. Res. Soc. Symp. Proc. 2006, 899E, N7.1. (43) Dimitrov, D. I.; Milchev, A.; Binder, K. Phys. ReV. Lett. 2007, 99, 054501. (44) Barrat, J. L.; Bocquet, L. Phys. ReV. E 1994, 49, 3079. (45) Bocquet, L.; Barrat, J. L. Phys. ReV. Lett. 1999, 82, 4671. (46) Levich, V. G. Physicochemical Hydrodynamics; Prentice-Hall: Englewood Cliffs, NJ, 1962. (47) Harden, J. L.; Cates M. E. Phys. ReV. E 1996, 53, 3782. (48) Harden J. L.; Borisov O. V.; Cates M. E. Macromolecules 1997, 30, 1179. (49) Webster M. A.; Yeomans J. M. J. Chem. Phys. 2005, 112, 164903. (50) Pastorino, C.; Binder, K; Kreer, T.; Mu¨ller, M. J. Chem. Phys. 2006, 124, 064902.

1234 Langmuir, Vol. 24, No. 4, 2008

DimitroV et al.

Figure 3. Change in the relative meniscus speed (a) and the total influx of fluid in the capillary (b) with growing chain length N of the brush coating and increasing wettability of the polymer brush pl. All data are normalized by data for a capillary without a coating.

Although it is difficult to vary the wall-liquid interaction experimentally and the resulting parameters entering eqs 1 and 2 in a continuous way, a practically useful way to modify wallliquid interactions is achieved by coating the wall with a polymer brush, as mentioned above. Despite their importance, however, investigations of the flow in brush-coated micro- and nanochannels are rather scarce. The deformations of grafted polymers in flows have so far been discussed47-49 for flat grafting surfaces only. Recently, an MD simulation by Tessier and Slater18 has studied the effect of end-grafted polymer chains on the electroosmotic flow in a cylindric nanotube. In a study of a polymer brush acting as a boundary of a flowing polymer melt composed of chains identical to those forming the brush, Pastorino et al.50 indeed found that the slip velocity strongly depends on the grafting density σG of the brush. Considering a polymer brush interacting with the particles of a simple liquid (acting as a solvent for the polymers), the effective wall-liquid surface tension and hence the contact angle θ can be varied.51 In previous work, we studied the structure of a dry brush grafted at the interior of cylindrical tubes in detail.54,55 The present work hence builds on our previous studies of capillary rise in nanotubes where the wall does not contain any grafted polymers43 and on the structure of brush-solvent systems51 and of brushes inside cylindrical pores54,55 by considering the capillary rise in a nanotube where the wall is coated with a polymer brush of intermediate grafting density. We vary the chain length and the wettability of the grafted polymers and consider the cases of both lyophilic and lyophobic (51) Dimitrov, D. I.; Milchev, A.; Binder, K. J. Chem. Phys. 2007, 127, 084905. (52) Allen M. P.; Tildesley D. J. Computer Simulation of Liquids; Clarendon Press: Oxford, 1987. (53) Soddermann, T.; Du¨nweg, B.; Kremer, K. Phys. ReV. E 2000, 68, 46702. (54) Dimitrov, D. I.; Milchev, A.; Binder, K. J. Chem. Phys. 2006, 125, 034905. (55) Dimitrov, D. I.; Milchev, A.; Binder, K.; Heermann, D. W. Macromol. Theory Simul. 2006, 15, 573.

Figure 4. Kinetics of total fluid uptake in a capillary, n2 ∝ Hz2 vs t, with attractive (a) and repulsive (b) walls for several degrees of brush coating wettability pl and a polymer length of N ) 24. The volume of liquid matter entering the tube is proportional to R2 × Hz ∝ n, where n is the number of liquid particles in the tube at time t and Hz denotes the height of the meniscus.

walls. In section II, the model is described, and the simulation technique is discussed. Section III presents our results for the time dependence of the fluid density profile in the capillary, the speed of the meniscus, and the resulting flux of the particles. In section IV, we discuss the flow profiles and the interplay of the flow with the resulting deformation and stretching of the polymer chains, and in section V, we summarize our results.

2. Model and Simulation Aspects The snapshot picture (Figure 1) illustrates our simulation geometry. We consider a cylindrical nanotube of radius R ) 10, whereby the capillary walls are represented by densely packed atoms forming a triangular lattice with lattice constant 1.0 in units of liquid atom diameter σ. Henceforth, all lengths will be quoted in units of σ. The wall atoms may fluctuate around their equilibrium positions at R + σ, subject to a finitely extensible nonlinear elastic (FENE) potential,

( )

UFENE ) -15wR02 ln 1 -

r2 , R0 ) 1.5 R02

(3)

Here w ) 1.0kBT, where kB denotes the Boltzmann constant and T is the temperature of the system. In addition, the wall atoms interact by a Lennard-Jones (LJ) potential

MD Simulations of Capillary Rise Experiments

ULJ(r) ) 4ww

Langmuir, Vol. 24, No. 4, 2008 1235

[( ) ( ) ] σww r

12

-

σww r

6

(4)

where ww ) 1.0 and σww ) 0.8. This choice of interactions guarantees no penetration of liquid particles through the wall while at the same time the mobility of the wall atoms corresponds to the system temperature. In all of our studies, we use a capillary length of Hmax ) 55. The top of the capillary is closed by a hypothetical impenetrable wall that prevents liquid atoms from escaping from the tube. At the bottom, the capillary is attached to a rectangular 40 × 40 reservoir for the liquid with periodic boundaries perpendicular to the tube axis; see Figure 1. Although the liquid particles may move freely between the reservoir and the capillary tube, initially, with the capillary walls and polymer chains being taken as distinctly lyophobic, these particles stay in the reservoir as a thick liquid film that sticks to the reservoir lyophilic upper wall. The liquid particles interact with one another by an LJ potential with ll ) 1.40 so that the resulting fluid attains a density of Fl ≈ 0.77. The liquid film is in equilibrium with its vapor both in the tube and in the partially empty right part of the reservoir. The brush coating of the inner wall of the tube consists of 156 end-grafted polymer chains containing N ) 8, 12, 16, 20, and 24 effective monomers each, at a grafting density of σG ) 0.05. The monomers at the anchoring end are bound with a FENE potential (eq 3) to virtual points that also form a triangular lattice. The LJ attraction between the liquid particles and the chain monomers is governed by an amplitude pl of varying strength 0 epl e2.2 that spans the whole range of brush wettability.

Figure 5. Advancement with time of the liquid density profile, measured around the point of its steepest descent, in the z direction along the tube axis for a capillary with repulsive (a) and with attractive (b) walls, coated by a wettable brush with chain lengths N ) 24 (a) and 8 (b).

Molecular dynamics (MD) simulations were performed using the standard velocity Verlet algorithm,52 carrying out typically 3 × 106 time steps with an integration time step of δt ) 0.01t0 where the MD time unit (tu) is t0 ) (σ2m/48LJ)1/2 ) 1/x48. (Note that the masses of both monomers and solvent particles were chosen to be equal at m ) 1.) As a check, runs with δt ) 0.0005t0 were also performed, but no significant differences could be detected. Temperature was held constant at T ) 1 using a standard dissipative particle dynamics (DPD) thermostat53 with a friction constant of ζ ) 0.5 and a step-function-like weight function with a cutoff of rc ) 1.5σpp. All interactions are cut off at rcut ) 2.5σ. At a time t ) 0, set to be equal to the onset of capillary filling, we turn the lyophobic wall-liquid interactions into lyophilic ones, and the fluid enters the tube. Then we perform measurements of the structural and kinetic properties of the imbibition process at equal time intervals. The capillary filling in brush-coated nanotubes is studied for two basic cases: (i) attractive (lyophilic) and (ii) repulsive (lyophobic) capillary walls whereby the wettability of the brush has been systematically varied each time. The brush-wall interaction has been kept neutral (nonadsorptive) in all cases. The total number of liquid particles is 25 000, and the number of particles forming the tube is 3243.

3. Capillary Filling Kinetics One of the main results of this study that describes the observed kinetics of capillary imbibition at varying degrees of wettability of the tube coating is shown in Figure 2 for the case of a nanotube with attractive walls. Despite fluctuations in some of the curves (which represent results from a single simulation run only), the data are in line with the Lucas-Washburn t1/2 relation of capillary rise with the time t elapsed after the onset of the imbibition process. The variation of the respective slopes of these curves with growing brush wettability pl, shown in the insets in Figure 2, indicates the rising speed of meniscus motion with the growing lyophilicity of the coating. One can readily see that the process of imbibition starts at a threshold value of pl ≈ 0.8. This is followed by a linear increase in imbibition “speed” (i.e., dH2(t)/dt) as the brush becomes more wettable up to pl ≈ 1.3. (Note that for this liquid a flat substrate turns from partially to completely wet at pl ≈ 1.4). Beyond that the speed slows down and tends toward saturation. Qualitatively, this behavior depends weakly on the chain length N as suggested by Figure 2a,b. A closer examination of the speed versus chain length relationship, however, reveals an interesting difference between meniscus “speed” and the total influx of liquid (per unit time) (cf. Figure 3). Evidently, the meniscus speed goes through a minimum at some intermediate chain length N ≈ 18, and the uptake of liquid by the capillary declines steadily as the polymer chains of the brush coating occupy progressive more and more of the inner part of the tube. Therefore, one may conclude that the observed velocity of meniscus motion is not necessarily representative of the efficiency of capillary imbibition in brushcoated nanotubes. It is also interesting to determine to what extent the degree of lyophilicity of the bare tube wall affects the imbibition process for differently wettable tube coatings. Somewhat surprisingly (Figure 4), one finds that capillary filling takes place even in nanotubes with lyophobic walls! Although Figure 4 suggests that the Lucas-Washburn t1/2 law appears to hold both for lyophilic and lyophobic capillaries, the similarity in the temporal evolution of the imbibition process in the two cases is, as a matter of fact, deceiving and the underlying

1236 Langmuir, Vol. 24, No. 4, 2008

DimitroV et al.

Figure 6. Velocity-field distribution around the moving edge of the fluid in a brush-coated nanotube: (a) N ) 8, lyophilic wall; (b) N ) 8, lyophobic wall; (c) N ) 24, lyophilic wall; and (d) N ) 24, lyophobic wall. Data are taken from the right half of the tube cross section with r ) 0 at the tube central axis and the tube wall being at r ) 10. All velocity fields are centered at the meniscus position, denoted by Z ) 0.

physical picture turns out to be quite different. Indeed, from the plot of the advancing front of the fluid with time t (Figure 5b) one can readily verify that in the case of a tube with attractive walls a front of steep increase in the liquid density turning into constant density at the meniscus position moves forward along the z axis. This density front retains its shape in time while its velocity declines, in line with the xt law of Lucas-Washburn. In contrast, in Figure 5b one may observe the liquid particles spreading through the polymer brush matrix although no region of constant liquid density behind any well-expressed interface (meniscus) can be detected. Thus, one might argue that in the lyophobic capillary the invading fluid simply soaks the wettable brush that coats and therefore neutralizes to some extent the repelling effect of tube walls by means of a kind of diffusive mechanism that is distinctly different from the curvature-driven capillary filling of Lucas-Washburn. In contrast to the case of nanotubes with wettable walls, it is thus not surprising that for lyophobic capillaries the polymer length of the brush coating plays a crucial role with respect to imbibition. These effects will be discussed in more detail in the next section. We end this brief description of the imbibition kinetics in brush-coated nanotubes by showing the velocity-field distribution

of the particles of the fluid in the vicinity of a moving meniscus during fluid uptake. Some distinct features in the atomistic picture of the fluid motion during imbibition in capillaries with wettable/ nonwettable walls and different lengths of the grafted polymers are immediately visible from Figure 6. For wettable capillaries and thin (N ) 8) brush coatings, fast hydrodynamics is observed mainly at and along the meniscus surface while the flow gradually slows down as one moves from the tube center to the tube wall. A thick (N ) 24) coating significantly kills the speed of the particles flowing through the polymer matrix, and the meniscus curvature is suppressed and the driving force is diminished. For liquid-repellent walls, in contrast, the thin coating swells, soaked by the fluid, and forms a quasi-wettable cylindrical wall parallel to the repulsive solid capillary wall, which supports a rising column of liquid even if no detectable meniscus if formed. The fluid transport proceeds predominantly through the middle of the tube. If the repulsive walls of the tube are coated by a thick brush reaching up to the tube center (N ) 24), one observes uniform, low speed, diffusive percolation of fluid particles through the brush matrix. Surprisingly, stronger flow is observed in the vicinity of the repulsive wall because the free volume immediately at the surface is larger.

MD Simulations of Capillary Rise Experiments

Figure 7. (a) Variation of the chain mean-squared end-to-end distance in the radial direction, 〈Rer2〉 (4), and along the tube axis, 〈Rez2〉 (O), within an interval of 10σ on both sides of the actual meniscus position. Above the plot, we give a schematic representation of a longitudinal capillary cross section with the meniscus position and flow direction indicated. Full symbols denote the same property in equilibrium. (b) Chain tilting along/against the flow in the vicinity of the moving meniscus. Positive (negative) projection measures the deviation with respect to the root monomer position of the grafted chain.

4. Flow Profiles and Structural Properties of the Polymer Coating In a recent study,54,55 we reported on the impact of cylindric confinement on the structural properties of a polymer brush in equilibrium. It is to be expected that the meniscus motion during capillary filling will strongly affect the structural properties of the polymer chains, especially in regions close to the meniscus position. One can clearly see from Figure 7a that the polymers immersed in the fluid immediately underneath the moving meniscus are in their maximum radially stretched state whereas those exposed to vapor above the meniscus are closer to the tube walls. Chains sufficiently far from the meniscus are more stretched in the radial direction if they are wetted by the fluid than those in the vapor phase. In the axial direction, however, chains are strongly stretched because of the flow far behind the meniscus and also immediately in front of it. Evidently, the latter sacrifice part of their conformational entropy in favor of energy gains by wetting parts of them close to the ends by the incoming fluid. Small deviations from the equilibrium data (full symbols in Figure

Langmuir, Vol. 24, No. 4, 2008 1237

Figure 8. (a) Density distribution of the polymer brush coating in the vicinity of a moving meniscus during capillary filling. Here, r denotes the distance from the tube center in the direction of the wall, N ) 16 and pl ) 2.0. The tube wall is indicated by densely packed spherical atoms. The area before the meniscus is seen to be completely empty of brush monomers whereas the monomer density is largest slightly behind the fluid interface. (b) Polymer bond orientation (arrows) of grafted chains around a moving meniscus. Gray-shaded areas indicate a different density of the invading fluid with the typical layering in the vicinity of the capillary wall.

7a) are solely due to the effect of flow behind the moving meniscus. (They are missing above the meniscus where the chains are exposed to the vapor phase.) Thus, because of different mechanisms, the polymer chains of the brush coating stretch considerably in the course of the capillary filling and also tilt (Figure 7b) differently depending on the meniscus actual position. It is evident that the axial projection of the end-to-end vector, Rez, undergoes a linear variation in the meniscus region of -20 ez e8 and changes sign at position z ≈ -7 in the meniscus wake. The polymer tilting leads to a unique effect characteristic of the brush-coating structure in the vicinity of the moving meniscussthe formation of a dense plug of monomers (Figure 8a), which assembles at the rear of the meniscus and trails after it in the process of fluid uptake. The orientation of bonds between neighboring monomers along the backbone of the polymer (Figure 8b) permits insight into this phenomenon of self-organization that pertains to capillary rise and should vanish in a tube with steady-state flow. Evidently, the running fluid stays out of the dense plug and close to the tube center (cf. the gray-shaded area in Figure 8b). In our simulations, we observe this plug formation in tubes with both attractive and repulsive walls, with the plugs being expressed at best for medium length chains of N ) 16. Of course, it is to be expected that the existence of such a dense

1238 Langmuir, Vol. 24, No. 4, 2008

DimitroV et al.

Figure 9. Variation of the total density of fluid particles (shaded red) and brush monomers (shaded blue) at the advancing meniscus: (a) N ) 24, lyophilic wall; (b) N ) 24, lyophobic wall; (c) N ) 8, lyophilic wall; and (d) N ) 8, lyophobic wall. The capillary axis is at r ) 0 and the wall is at r ) 10 so that only half of the density distribution across the tube is shown. The z coordinate measures the distance ahead and behind the meniscus position. The surface layering and fluid-brush segregation effects are well pronounced for wettable walls and long N ) 24 polymers. For lyophobic walls and a thick coating (N ) 24), a meniscus with negative curvature is formed.

moving plug of monomers will influence the hydrodynamics of the fluid in the immediate vicinity of the moving interface. Indeed, we find that the overall speed of capillary rise is reduced when a plug follows the meniscus motion. It could be that this effect explains the observed minimum in the speed-chain length relationship, displayed in Figure 3a. In ending this section, we point out that the complex interplay between liquid-liquid, liquid-polymer, and liquid-wall interactions leads to pronounced effects of fluid-polymer segregation in the vicinity of the capillary wall, as shown in Figure 9. It becomes clear from Figure 9a that the combined effect of wall attraction (wetting) on the fluid particles and repulsion, acting on the brush monomers, along with the topological connectivity of the grafted macromolecules leads to the formation of alternating layers, occupied predominantly by either liquid or polymer particles. These layers run parallel to the tube walls and become less well expressed as the length of the grafted chains becomes shorter (Figure 9c). For lyophobic walls, in contrast, no layering whatsoever is observed. The fluid uptake proceeds by a kind of invasive percolation through the wettable polymer matrix that serves rather as a porous medium (Figure 9b) if the chains of the coating reach sufficiently far so as to occupy the tube center. For shorter chains, however, the wetted coating provides an attractive concentric cylindric wall “screening” the nonwettable solid capillary wall (Figure 9d). In fact, this wettable concentric inner wall provides additional support and thus helps the fluid to form a meniscus that then runs in an effectively narrower tube. One should also point out at the end of this section that the aforementioned results have been derived by extensive MD simulations that require considerable computational effort. It is therefore appropriate to mention at this point as a possible

alternative the recently developed lattice Boltzmann method56-58 that has proved rather efficient for modeling a number of flow problems in channels and capillaries. It has been demonstrated59 that this mesoscopic method yields similar results to MD at least for the flow of simple liquids around obstacles if the relevant parameters are appropriately chosen. In the present case of capillary flow in a tube with a deformable brush coating, however, possible modeling within the lattice Boltzmann approach is rather difficult and therefore remains outside the scope of this study.

5. Summary and Conclusions We have carried out extensive MD simulations of capillary filling in a nanotube, coated inside by a polymer brush whereby both the wettability of the tube walls and the brush coating and its thickness were systematically varied. The results of this investigation display rich hydrodynamic behavior depending on the brush/fluid interaction and the wettability of the capillary wall. • We find that capillary filling takes place in both wettable/ nonwettable tubes, provided the polymer brush coating of the tube wall is wetted by the fluid. • The capillary rise always proceeds by a t1/2 law whereas the governing mechanism behind it (Lucas-Washburn or diffusive propagation) may be different, depending on chain length N and the wettability of the tube walls. (56) Benzi, R.; Succi, S.; Vergassola, M. Phys. Rep. 1992, 222 145. (57) Gladrow, D. W. Lattice Gas Cellular Automata and Lattice Boltzmann Models; Lecture Notes in Mathematics; Springer: Berlin, 2000. (58) Succi, S. The Lattice Boltzmann equation; Oxford University Press: Oxford, 2001. (59) Horbach, J.; Succi, S. Phys. ReV. Lett. 2006, 96, 224503.

MD Simulations of Capillary Rise Experiments

• For a wettable wall of the nanotube, the speed of fluid imbibition drops steadily with growing chain length N whereas the meniscus speed goes through a minimum at intermediate chain lengths. During fluid uptake, the polymer brush reorganizes into alternating layers of changing monomer density in the course of fluid-monomer segregation. A dense plug of monomers forms under the meniscus, following its motion in the capillary. • For a nonwettable capillary wall (coated by a wettable polymer brush), one finds three different regimes of capillary rise, depending on polymer chain length N: ο Short chains: no meniscus motion but an influx of fluid percolating diffusively through the wet brush coating. ο Intermediately long chains: coaxial regions of low monomer density permeable to the fluid, segregate inside the brush by

Langmuir, Vol. 24, No. 4, 2008 1239

percolative spreading whereby a meniscus is formed, which then moves further along the tube. ο Long chains: A fluid front (meniscus of nearly vanishing curvature) rises up the capillary by a mechanism such as imbibition through a porous medium. Acknowledgment. D.I.D. is grateful for support from the Max Planck Institute of Polymer Research via an MPG fellowship, and A.M. received partial support from the Deutsche Forschungsgemeinschaft (DFG) under project no. 436BUL113/130. A.M. and D.I.D. were supported by project “INFLUS”, NMP031980 of the VI-th FW program of the EC. LA7019445