Langmuir 2006, 22, 6739-6742
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Designing a Simple Ratcheting System to Sort Microcapsules by Mechanical Properties Kurt A. Smith, Alexander Alexeev, Rolf Verberg, and Anna C. Balazs* Department of Chemical & Petroleum Engineering, UniVersity of Pittsburgh, Pittsburgh, PennsylVania 15261 ReceiVed April 13, 2006. In Final Form: June 14, 2006 Using computational modeling, we analyze the fluid-driven motion of compliant particles over a rigid, saw-toothed surface. The particles are modeled as fluid-filled elastic shells and, thus, simulate ex vivo biological cells or polymeric microcapsules. Through the model, we demonstrate how the patterned surface and an oscillatory shear flow can be combined to produce a ratcheting motion, yielding a straightforward method for sorting these capsules by their relative stiffness. Since the approach exploits the capsule’s inherent response to the substrate, it does not involve explicit measurement and assessment. Because the process utilizes an oscillatory shear, the sorting can be accomplished over a relatively short portion of the substrate. Due to these factors, this sorting mechanism can prove to be both efficient and relatively low-cost.
Establishing facile, noninvasive methods for sorting mixtures of micron-scale particles by their mechanical properties remains a challenging task and one that is vitally important for biotechnology, as well as colloid science. In particular, there are a number of diseases that alter the elasticity of biological cells1 and simple techniques that could separate normal from diseased samples would facilitate further studies. In addition, scientists are now generating new types of polymeric microcapsules2 and “polymersomes”,3 with a range of tailored compositions and flexibilities. For instance, relatively rigid capsules can be produced by incorporating nanoparticles into the bounding shells,4 whereas more elastic species can be formed from block copolymers.3 To utilize these polymeric capsules as robust microcarriers or microreactors, it becomes necessary to isolate species with the desired mechanical properties. Using computational modeling, we analyze the fluid-driven motion of compliant particles over a rigid substrate. The particles are modeled as fluid-filled elastic shells and thus simulate ex vivo biological cells or polymeric microcapsules. Through the model, we demonstrate how a ratchetlike surface and an oscillatory shear flow can be combined to yield a straightforward approach for sorting these capsules by their relative stiffness. Although researchers have recently harnessed ratcheting mechanisms for separating microparticles by size5 and optical behavior,6 this study is the first to describe * To whom correspondence
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(1) Suresh, S.; Spatz, J.; Mills, J. P.; Micoulet, A.; Dao, M.; Lim, C. T.; Beil, M.; Seufferlein, T. Acta Biomater. 2005, 1, 15-30. (b) Van Vliet, K. J.; Bao, G.; Suresh, S. Acta Mater. 2003, 51, 5881-5905. (2) See, for example: (a) Gao, C. Y.; Leporatti, S.; Moya, S.; Donath, E.; Mo¨hwald, H. Langmuir 2001, 17, 3491-3495. (b) Lulevich, V. V.; Radtchenko, I. L.; Sukhorukov, G. B.; Vinogradova, O. I. Macromolecules 2003, 36, 28322837. (c) Fery, A.; Dubreuil, F.; Mohwald, H. New J. Phys. 2004, 6, 18. (d) Vinogradova, O. I. J. Phys.: Condens. Matter 2004, 16, R1105-R1134. (e) Peyratout, C. S.; Dahne, L. Angew. Chem., Int. Ed. 2004, 43, 3762-3783. (3) Discher, B. M.; Won, Y.-Y.; Ege, D. S.; Lee, J. C.-M.; Bates, F. S.; Discher, D. E.; Hammer, D. A. Science 1999, 284, 1143-1146. (b) Discher, D. E.; Eisenberg, A. Science 2002, 297, 967-973 and references therein. (4) Skirtach, A. G.; Dejugnat, C.; Braun D.; Susha, A. S.; Rogach, A. L.; Parak, W. J.; Mohwald, H.; Sukhorukov, G. B. Nano Lett. 2005, 5, 1371-1377. (5) See for example: (a) Savel’ev, S.; Marchesoni, F.; Nori, F. Phys. ReV. E 2005, 71, 011107. (b) Marquet, C.; Buguin, A.; Talini, L.; Silberzan, P. Phys. ReV. Lett. 2002, 88, 168301. (c) Matthias, S.; Mu¨ller, F. Nature 2003, 424, 53-57. (d) Dere´nyi, I; Astumian, R. D. Phys. ReV. E 1998, 58, 7781-7784. (e) van Oudenaarden, A.; Boxer, S. G. Science 1999, 285, 1046-1048. (f) Cabodi, M.; Chen, Y.-F.; Turner, S. W.; Craighead, H. G.; Austin, R. H. Electrophoresis 2002, 23, 3496-3503. (g) Slater, G.; et al. Electrophoresis 2002, 23, 3791-3816. (6) Feringa, B. L.; et al. Appl. Phys. A 2002, 75, 301-308.
how an asymmetric surface in a zero-average field can be utilized to discriminate among particles of varying modulus. Since the approach exploits the particle’s inherent response to the substrate, it does not involve explicit measurement and assessment and thus could prove to be a highly efficient method for carrying out the separation process. Figure 1 shows the graphical output from our simulations for two mechanically distinct, fluid-filled capsules, which are driven to move along the substrate by an externally imposed, oscillatory flow. To accurately capture the behavior of the system, it is necessary to model the interactions among the external fluid, the elastic shell, the encapsulated fluid, and the substrate. To this end, we couple7,8 the lattice Boltzmann model (LBM),9 which is an efficient solver for the Navier-Stokes equation, with the lattice spring model (LSM),10 which is used to simulate the continuum equations for an elastic solid. Through boundary conditions at the fluid-solid interface,11 the fluids exert forces on the capsule’s compliant shell and, in turn, the shell “pushes back” on the fluids. In this manner, we capture the complex fluid-structure interactions within the system. In these two-dimensional simulations, the thin shell of the capsule is modeled by a cylindrically symmetric lattice of harmonic springs, consisting of three concentric layers of N ) 120 nodes that lay a distance ∆r apart. We use eight springs to connect each node to its nearest and next-nearest neighbors. By varying the spring constant, k, we alter the shell’s Young’s modulus, E ) 5k/2∆x, where ∆x is the equilibrium separation between the nodes. The dynamic behavior of the LSM nodes is governed by Newton’s equation of motion. The LBM for the fluid consists of two processes: the first being the propagation of fluid “particles” to neighboring lattice sites and the second being collisions between particles when they reach a site. The fluid particles are representative of mesoscopic portions of the fluid and are characterized by a particle (7) Alexeev, A.; Verberg, R.; Balazs, A. C. Macromolecules 2005, 38, 1024410260. (8) Buxton, G. A.; Verberg, R.; Jasnow, D.; Balazs, A. C. Phys. ReV. E 2005, 71, 056707-056724. (9) Succi, S. The Lattice Boltzmann Equation for Fluid Dynamics and Beyond; Clarendon Press: Oxford, 2001. (10) Ladd, A. J. C.; Kinney, J. H.; Breunig, T. M. Phys. ReV. E 1997, 55, 3271-3275. (d) Ladd A. J. C.; Kinney, J. H. Physica A 1997, 240, 349-360. (11) Bouzidi, M.; Firdaouss, M.; Lallemand, P. Phys. Fluids 2001, 13, 34523459.
10.1021/la0610093 CCC: $33.50 © 2006 American Chemical Society Published on Web 07/08/2006
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Figure 1. Representative images of elastic capsules in reverse (left) and forward (right) shear flow. Images a and c are for a stiff capsule with Ca ) 1.2 × 10-3 and Φ ) 0.58. Images b and d are for a soft capsule with Ca ) 9.6 × 10-3 and Φ ) 4.6. The arrow in each image indicates the direction of the imposed shear. The value of the local strain in the capsule’s shell is indicated by the color bar at the bottom of the figure. These images show only a section of the surface. In these examples, the angle of incline of the substrate is 8.5°.
distribution function, which describes the mass density of fluid particles as a function of position, time, and particle velocity. The hydrodynamic quantities of interest are then obtained as moments of this distribution function. The simulation proceeds through the sequential update of both the lattice spring and the lattice Boltzmann systems.7,8 The LSM system is updated by first calculating the forces that act on the LSM nodes, due to the LSM springs, the enclosed fluid, the surrounding solvent, and interactions with the substrate. New positions, velocities, and accelerations of the LSM nodes are then calculated using the Verlet algorithm.12 In updating the LBM system, we first establish which LBM links intersect the solid/fluid interface. We then obtain the velocities at these points of intersection from neighboring LSM nodes. Next, we propagate the distribution function by moving fluid particles to their neighboring nodes whenever these nodes are in the fluid domain, and otherwise, we apply the appropriate boundary condition. Finally, we modify the distribution functions at the LBM nodes to account for the collision step. We then repeat the entire cycle. Within our simulation box, the bottom, stationary wall has a saw-toothed pattern, which gives rise to the ratcheting behavior described further below. We examine two substrates, one where the angle of incline is 6° and another where it is 8.5°. The box length in the flow direction (the x direction) is Lx ) 500, whereas the height between the top of the substrate and the upper wall is Ly ) 70. Periodic boundary conditions are applied along x. From our prior studies on capsule-capsule interactions,13 we found this box length to be sufficient in order to avoid effects from periodic images. Oscillatory shear, with period τ, is introduced by moving the flat, upper wall of the simulation box with a constant velocity of +Uwall for a time of τ/2 and -Uwall for a subsequent time of τ/2. Note that the oscillating flow is symmetric and does not yield a net displacement of the fluid. Herein, we focus on a single capsule that interacts with the sloped portions of the substrate through a nonspecific interaction, modeled by a Morse potential, φ(r) ) (1 - exp[-((r - r0)/ κ)])2, where and κ characterize the respective strength and range of the interaction potential, r is the distance between a LSM node on the shell’s outer surface and a site on the substrate, and r0 is that distance between the capsule and the substrate where the force is equal to zero. There is no attraction between the capsule and the vertical portions of the saw-tooth. Substrates (12) Tuckerman, M.; Berne, B. J.; Martyna, G. J. J. Chem. Phys. 1992, 97, 1990-2001. (13) Alexeev A.; Verberg R.; Balazs A. C. Soft Matter 2006, 2, 499-509.
with such features can be prepared by chemical patterning techniques that permit preferential deposition of an attractive coating on the sloped portions of the surface.14,15 We can characterize the motion of the capsule in the flowing fluid by the Reynolds number Re ) Fγ˘ R2/µ where F and µ are the respective fluid density and viscosity of an undeformed capsule, R is the undeformed capsule radius, and γ˘ ) Uwall/Ly is the shear rate. We operate in the Stokes regime where Re < 1. Given that E is the Young’s modulus of the capsule shell, and h is the shell thickness, then the capillary number can be written as Ca ) γ˘ µR/Eh, which represents the strength of viscous shear stress relative to the elastic stress in the capsule’s shell. We also define a dimensionless interaction strength Φ ) N/Ehκ2, which represents the ratio of the adhesion strength to the capsule stiffness and is independent of flow. Our results cover a range of Ca ≈ 10-3-10-2 and Φ ≈ 0.5-5. Figure 1a is a snapshot of a relatively stiff capsule; when this capsule is driven by the imposed shear in the reverse (-Uwall) direction, it cannot surmount a ridge in the substrate and becomes stuck. In contrast, a softer, more deformable capsule (Figure 1b) can be driven to ascend the ridge and, hence, move backward. This difference in the capsules’ responses to the barrier has significant consequences. When the direction of the shear changes, the stiff capsule easily moves forward one step along the substrate (Figure 1c); in fact, it continually advances along the surface because in an oscillating shear, the ratchet-like surface serves to rectify the motion of the stiff species. The softer capsule, however, has “lost ground” by moving backward and therefore, advances more slowly (Figure 1d). If the flow period is sufficiently short, the soft capsule can even spend its entire time oscillating between configurations similar to those in panels b and d in Figure 1, without ever advancing along the surface. A more quantitative description of the capsule dynamics is shown in Figure 2, which displays distance (along x) versus time for capsules of different E, in an oscillating shear flow. As can be seen, the stiffer capsules move ahead, whereas much softer ones will essentially oscillate between two locations (at x ≈ 0.6 and x ≈ 1.9). In the case of the stiffer capsules, the plateaus in the plot indicate points where the particle is lodged against the ridge, as in Figure 1a. (14) Rockford, L.; Liu, Y.; Mansky, P.; Russell, T. P.; Yoon, M.; Mochrie, S. G. J. Phys. ReV. Lett. 1999, 82, 2602-2605. (15) We note that by making the vertical portion of the substrate nonattractive, we can reduce the tendency of a capsule to become stuck in a corner and thus increase the regime over which ratcheting is effective.
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Figure 2. Capsule position, x, vs time, t, in oscillatory shear flow for capsules of different stiffness. Time is expressed in units of the flow period. (One flow period is equal to 104 000 lattice Boltzmann time steps.) Capsule position is expressed in units equal to the length of one saw tooth on the substrate. Results, in order of decreasing capsule stiffness, are for Ca ) 1.2 × 10-3, Φ ) 0.58 (black solid line), Ca ) 4.8 × 10-3, Φ ) 2.3 (red dashed line), and Ca ) 9.6 × 10-3, Φ ) 4.6 (blue dotted line). The angle of incline of the substrate is 8.5°.
Figure 3. Superposition of capsules of differing stiffness after 1.5 periods of oscillatory shear flow. Each colored capsule represents a separate simulation; the capsule at the far left indicates the initial position in all cases. Capsule colors correspond to the following cases, in order of decreasing stiffness: red (rigid capsule), blue (Ca ) 1.2 × 10-3, Φ ) 0.58), green (Ca ) 2.4 × 10-3, Φ ) 1.16), yellow (Ca ) 4.8 × 10-3, Φ ) 2.3), and violet (Ca ) 9.6 × 10-3, Φ ) 4.6). The angle of incline of the substrate is 8.5 degrees.
An important consequence of the above behavior is that after a fixed time interval, ∆t, the location of different capsules on the substrate will depend on their relative stiffness. Figure 3 shows a superposition of simulations for capsules of different stiffness that were run for a fixed time of 1.5 periods of the oscillating shear flow. The capsules are arrayed on the surface with the stiffest capsule located in the lead, the next stiffest in second place, and so on. In effect, the capsules sort themselves, as a consequence of their different responses to the substrate. To facilitate the design of the corresponding experiments, we express our simulation parameters in terms of physical values by considering our particles to represent 50 µm polymeric capsules, such as those fabricated through the layer-by-layer deposition process,16 in an aqueous solution, for which µ ≈ 10-3 kg/(m‚s) and F ≈ 103 kg/m3. These values set the length and time scales for our simulation, so that a lattice Boltzmann unit is equivalent to 1 µm in length and one time step is equivalent to 0.17 µs. Given that N is the number of nodes in a layer of the capsule’s shell and is strength of the interaction in the Morse potential (see above), then N is a measure of the energy of adhesion between the capsule and the surface. We choose parameters such that the N ≈ 10-14J,17 and the interaction range (16) Dubreuil, F.; Elsner, N.; Fery, A. Eur. Phys. J. E 2003, 12, 215-221. (17) Nolte, M.; Fery, A. Langmuir 2004, 20, 2995-2998.
Figure 4. (a) Phase diagram showing capsule behavior in long period oscillating shear as a function of inverse capsule stiffness (Eh)-1 and shear rate γ˘ . (Eh)-1) 0 corresponds to a rigid capsule. Four regimes are observed: capsule is stuck (red crosses), ratcheting (green circles), overdriven (blue squares), and reverse ratcheting (black triangles). The angle of incline of the substrate is 6°. (b) Average capsule velocity as a function of shear rate in long period oscillating shear. Results are for rigid capsules (blue plusses) and capsules with stiffness of 0.075 Pa‚m (red crosses), 0.038 Pa‚m (green squares), 0.019 Pa‚m (black diamonds), 0.015 Pa‚m (blue dots), and 0.01 Pa‚m (purple triangles).
is κ ≈ 0.1 µm,18 which characterizes an electrostatic interaction between a microcapsule and an oppositely charged surface. We examine capsules with stiffness Eh in the range of 0.01-0.075 Pa‚m. Such values are attainable via polyelectrolyte multilayer assembly.19 Finally, we consider typical experimental values of the shear rates on the order of γ˘ ≈ 103 s-1. Using these values, we construct the plots described below. In addition, to provide a robust quantitative analysis, we determine the capsule dynamics in the limit of long period flows (since for small values of τ, the system is sensitive to transient effects). We do this by measuring the steady-state capsule velocity in the forward flow, the same in reverse flow, and taking the difference between the two to obtain the average capsule velocity in the oscillatory shear. Focusing on these long period flows, we generate the phase map in Figure 4a, which reveals the four regimes that we observed as a function of the capsule stiffness, Eh, and shear rate, γ˘ . At low γ˘ , most capsules are stuck in a given step on the substrate, as neither forward nor reverse shear is sufficiently high to move the capsule to the next step; this regime is marked by red crosses in the plot. At a critical value of γ˘ , the viscous force becomes (18) Bosio, V.; Dubreuil, F.; Bogdanovic, G.; Fery, A. Colloids Surf. A: Physicochem. Eng. Aspects 2004, 243, 147-155. (19) Thompson, M. T.; Berg M. C.; Tobias I. S.; Rubner M. F.; Van Vliet, K. J. Biomaterials 2005, 26, 6836-6845.
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sufficiently large to push the capsule over a ramp in forward shear but not large enough to push it backward in reverse shear. In this regime, the capsule’s motion is rectified by the ratcheting surface, and hence, it moves forward step by step along the substrate. This “ratcheting regime” is marked by green dots. As γ˘ is increased further, the capsule is eventually able to move backward across a ramp while under reverse shear. In this “overdriven” regime (shown in blue squares), the capsule moves alternately forward and backward. Its velocity is still positive but is greatly reduced. The softest capsule exhibits a “reverse ratcheting” (marked by black triangles), because the critical value of γ˘ needed to move the capsule one step backward is less than the critical value needed to move it one step forward. Between these critical γ˘ values, the capsule velocity is negative (see Figure 4b). When a capsule moves forward over a ridge, it temporarily detaches from the substrate and, thus, requires sufficient force to overcome the adhesive interaction with the surface. The adhesion energy does not depend strongly on Eh. In contrast, a capsule moving backward over a ridge stays in contact with the substrate the entire time. The energy barrier, in this case, is mainly due to the energetic cost of bending the capsule around the ridge and is roughly proportional to Eh. Thus, as Eh is decreased, one can expect a crossover into a regime where less force is needed to move the capsule backward than to move it forward. In Figure 4b, we plot the average capsule velocity versus shear rate γ˘ for representative points. The figure illustrates the large differences in capsule velocity across the different regimes described above. For example, a capsule with stiffness Eh ) 0.075 Pa‚m (see red crosses) remains stuck below γ˘ ≈ 1300 s-1 and has zero velocity. When the shear rate is increased to γ˘ ≈ 1350 s-1, the system enters the ratchet regime and the capsule velocity not only increases sharply but also continues to increase approximately linearly with γ˘ for the duration of the ratchet regime. At γ˘ ≈ 1800 s - 1, the system enters the overdriven regime, where the capsule moves backward when the reverse shear is applied. As a result, the average capsule velocity decreases dramatically. The large differences in velocities at fixed γ˘ point to a simple procedure for sorting capsules by stiffness. In particular, one could initially induce flow with γ˘ ≈ 2000 s-1, where rigid capsules advance approximately five times faster than capsules with Eh ) 0.075 Pa‚m. After removing the most rigid capsules, one could then reduce the shear rate to γ˘ ≈ 1750 s-1. At this value, capsules with Eh g 0.075 Pa‚m move much faster than softer capsules with Eh e 0.038 Pa‚m (see Figure 4b). By incrementally reducing γ˘ , one could sort a quantity of capsules into a series of wellseparated groups, where each group contains only capsules whose stiffness lies in a certain range. Because of the large difference between the velocities of “ratcheting” capsules and “overdriven” capsules, this sorting could be accomplished over a short track length. The selectivity of this system with respect to capsule stiffness can be explained in terms of a simple model for the over-damped motion of a particle, at position x ) (x, y), in a spatially periodic but asymmetric potential field U(x), experiencing a symmetric, temporally periodic forcing F(t). The particle motion is described by the equation ∂x/∂t ) -∇U + F(t). In Figure 5, for both a soft and stiff capsule, we plot U(x), which is the sum of the capsule’s elastic energy (obtained from the LSM springs) and interaction energy with the substrate (obtained from the Morse potential). To calculate this plot, we fix the position x of the capsule’s center of mass and allow the capsule to reach its equilibrium structure in the absence of shear. Figure 5b shows that there is
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Figure 5. Energy landscapes for capsules in the vicinity of a ridge. (a) Soft capsule (Φ ) 4.6 or Eh ) 0.01 Pa‚m). (b) Stiff capsule (Φ ) 0.58 or Eh ) 0.075 Pa‚m). The color bar indicates the value of the total energy, which is the sum of the capsule’s elastic energy (obtained from the LSM springs) and interaction energy with the substrate (obtained from the Morse potential). The angle of incline of the substrate is 8.5°.
a steep energy barrier for a stiff capsule to cross a ridge when moving backward; this is due to the high-energy cost of deforming the capsule around the ridge. Because of this energetic cost, the substrate creates an effective ratcheting potential for a stiff capsule. The situation, however, is quite different for the soft capsule in Figure 5a. The asymmetry of the energy landscape is much less pronounced and there is a relatively low energy pathway for the capsule to cross the ridge while moving backward. Hence, the substrate is much less effective as a ratchet for the soft capsule. By overlaying the trajectories of the moving particles on these energy landscapes, we confirm that the capsules move through low energy regions of space, and that the stiff capsule becomes stuck in the vicinity of the large energy gradient. Thus, it is the capsule’s inherent mechanical properties that dictate its response to the surface and allow this simple system to be used for separations. In summary, we used computer simulations to design a ratcheting system that can be harnessed to separate elastic particles (e.g., biological cells or microcapsules) by their mechanical properties. Since the method utilizes oscillatory shear, the particle separation can be accomplished over a relatively small surface area and, thus, could be readily incorporated into “lab-on-a chip” devices. Acknowledgment. The authors gratefully acknowledge discussions with Dr. V. V. Yashin and funding from DOE (to A.A.), ONR (to R.V.), and NSF (to K.A.S). LA0610093
