Langmuir 2007, 23, 983-987
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Patterned Surfaces Segregate Compliant Microcapsules Alexander Alexeev, Rolf Verberg, and Anna C. Balazs* Chemical Engineering Department, UniVersity of Pittsburgh, Pittsburgh, PennsylVania 15261 ReceiVed October 4, 2006. In Final Form: December 19, 2006 For both biological cells and synthetic microcapsules, mechanical stiffness is a key parameter since it can reveal the presence of disease in the former case and the quality of the fabricated product in the latter case. To date, however, assessing the mechanical properties of such micron-scale particles in an efficient, cost-effective means remains a critical challenge. By developing a three-dimensional computational model of fluid-filled, elastic spheres rolling on substrates patterned with diagonal stripes, we demonstrate a useful method for separating cells or microcapsules by their compliance. In particular, we examine the fluid-driven motion of these capsules over a hard adhesive surface that contains soft stripes or a weakly adhesive surface that contains “sticky” stripes. As a result of their inherently different interactions with the heterogeneous substrate, particles with dissimilar stiffness are dispersed to distinct lateral locations on the surface. Since mechanically and chemically patterned surfaces can be readily fabricated through soft lithography and can easily be incorporated into microfluidic devices, our results point to a facile method for carrying out continuous “on the fly” separation processes.
There are a number of diseases (e.g., malaria and various cancers) that alter the elasticity of biological cells,1 and, in some instances, the stiffness of the cell indicates the stage of infection.2 While researchers are currently developing sophisticated approaches (involving optical tweezers) to measure the mechanical characteristics of diseased cells,3 there remains a critical need for facile methods to sort cells by their stiffness, and thereby enable rapid, low-cost assays. In addition, scientists are now generating new types of polymeric microcapsules4 and “polymersomes”,5 with a range of tailored compositions and flexibilities. For instance, relatively rigid capsules are produced by incorporating nanoparticles into the bounding shells,6 while more elastic species are formed using block copolymers.5 In order to utilize these polymeric capsules as robust microcarriers or microreactors, it becomes necessary to isolate species with the desired mechanical properties. Using a novel computational approach, we show how surfaces that exhibit a simple mechanical or chemical variation can be harnessed to effectively separate microcapsules according to their compliance. The microcapsules are modeled as three-dimensional fluid-filled elastic shells and thus simulate ex vivo biological cells or polymeric capsules.4 Through an imposed flow, these compliant capsules are driven to move over a hard surface that contains soft stripes (Figure 1), or a weakly adhesive surface that contains “sticky” stripes (Figure 2). As we show below, a capsule’s path on these heterogeneous substrates is highly dependent on its stiffness; that is, mechanically different particles that initially start at the same position eventually roll to separate locations on the substrate, and thus are effectively sorted by the surface. Since this 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 continuous, “on-the-fly” separation processes. Furthermore, since recent advances in soft * To whom correspondence should be addressed. E-mail: balazs1@ engr.pitt.edu. (1) Suresh, S. J. Mater. Res. 2006, 21, 1871. (2) Suresh, S.; Spatz, J.; Mills, J. P.; Micoulet, A.; Dao, M.; Lim, C. T.; Beil, M.; Sefferlein, T. Acta Biomater. 2005, 1, 15. (3) Van Vliet, K. J.; Bao, G.; Suresh, S. Acta Mater. 2003, 51, 5881. (4) Fery, A.; Dubreuil, F.; Mohwald, H. New. J. Phys. 2004, 6, 18. (5) 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. (6) Shchukin, D. G.; Sukhorukov, G. B.; Mohwald, H. Angew. Chem., Int. Ed. 2003, 42, 4472.
Figure 1. Snapshots of an elastic capsule rolling along an adhesive, mechanically patterned substrate in shear flow. The system is shown in cross-section (made through the capsule’s center of mass). The colors in the capsule and substrate reveal the strain (see color bar). The arrows indicate the flow direction, and the arrow’s color indicates the magnitude of the velocity. The lines on the substrate mark the boundaries of a soft patch. The capsule approaches (upper panel) and moves past (lower panel) a soft patch. Note the strong deformation of the soft patch due to the adhesive interaction with the capsule and, consequently, the increase in contact area between the capsule and substrate.
lithography7 make it possible to readily fabricate such mechanically or chemically patterned surfaces, the approach can provide a relatively cost-effective method for carrying out the assays. To capture the complex dynamic interactions between the fluids and surfaces in this system, we integrate the lattice (7) Chen, C. S.; Jiang, X. Y.; Whitesides, G. M. MRS Bull. 2005, 30, 194.
10.1021/la062914q CCC: $37.00 © 2007 American Chemical Society Published on Web 01/06/2007
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Figure 2. Snapshots of an elastic capsule rolling along a rigid substrate patterned with adhesive strips in shear flow. The system is shown in cross-section (made through the capsule’s center of mass). The lines on the substrate indicate the boundaries of a sticky patch. The characteristics of the strain and velocity are depicted in the same manner as in Figure 1. The capsule is located in front of a sticky patch (upper panel) and is crossing a sticky patch (lower panel). Note the capsule’s flattening due to a stronger adhesion to the sticky patch. The capsule’s deformation enhances the contact area with the substrate.
Boltzmann model (LBM) for hydrodynamics and the lattice spring model (LSM) for the micromechanics of elastic solids. More specifically, the LBM is an efficient solver for the NavierStokes equations.8 In the LSM, a compliant material is described by a network of harmonic “springs”, which interconnect nearest and next-nearest neighboring lattice nodes. The two systems interact through appropriate boundary conditions.9,10 In particular, the velocities of lattice spring nodes situated at the solid-fluid interface are transmitted to the surrounding fluids. In turn, these LSM nodes experience forces due to the fluid pressure and viscous stresses at the interface; this force is applied as a load to the neighboring LSM nodes. The total force on the LSM nodes also includes the spring force between nodes and an adhesion force (described further below). To model the substrate, we use a simple cubic lattice, where springs of stiffness k connect the nearest and next-nearest neighbor LSM nodes. The Young’s modulus of the solids is given by E ) 5k/2∆xLS, the Poisson’s ratio is ν ) 1/4, and the solid density is Fs ) M/∆xLS3, where ∆xLS is the lattice spacing in the LSM, and M is the mass at each node. To mimic the mechanical patterning of the surface, we vary k along the substrate. The capsule’s three-dimensional shell is constructed from two concentric layers of LSM nodes. By using the Delaunay triangulation technique,11 we distribute nodes in a regular manner on each surface. These two concentric surfaces are separated by (8) Succi, S. The Lattice Boltzmann Equation for Fluid Dynamics and Beyond; Clarendon Press, Oxford University Press: Oxford, New York, 2001. (9) Alexeev, A.; Verberg, R.; Balazs, A. C. Phys. ReV. Lett. 2006, 96, 148103. (10) Alexeev, A.; Verberg, R.; Balazs, A. C. Macromolecules 2005, 38, 10244. (11) Delaunay, B. N. Bull. Acad. Sci. USSR: Cl. Sci. Math. 1934, 7, 793.
Letters
a distance that is equal to the average size of a triangular bond and are connected by springs between the nearest and nextnearest neighbor nodes. The spring constant for springs located on the capsule surfaces and normal to the surfaces is k, while the diagonal springs have spring constants of 2/3k. The dimensions of our simulation box (in LBM units) are length Lx ) 120, height Ly ) 40, and width Lz ) 60 (see Figures 1 and 2). The top wall (at y ) Ly) moves with a constant velocity (U0, 0, 0) and serves to drive the flow; periodic boundary conditions are applied in the x and z directions. We set the density and viscosity of both the host and the encapsulated fluids to Ff ) 1 and µ ) 1/6, respectively. The stationary, bottom wall (at y ) 0) represents the floor of our substrate. The spacing between the LSM nodes in the substrate is ∆xs ) 1.25; the substrate thickness is hs ) 10, and density is Fs ) 1.5. The outside radius of the undeformed capsule is R ) 10, and the capsule density is Fc ) 1. The number of LSM nodes on this outer radius is Nc ) 642; the average spacing between nodes is ∆xc ≈ 1.4, and the distance between the outer and inner layers (thickness of the capsule’s shell) is h ) 1.4.12 The following Morse potential describes the capsule-substrate interaction: φ(r) ) (1 - exp[-(r - r0)/κ])2. Here, and κ ) 1 characterize the respective strength and range of the interaction potential, r is the distance between a pair of LSM nodes, where one node lies on the capsule’s outer surface and the other lies at the substrate-fluid interface, and r0 ) 1.8 is the equilibrium distance where the force due to the potential is zero. To prevent overlap between the capsule and the substrate, we fix ) r ) 0.005 for the repulsive part of the potential (r < r0), while the values for the adhesive part (r > r0) ) a are specified below. The flow velocity is such that the Reynolds number Re , 1, meaning that inertial effects can be neglected compared to viscous effects. To characterize the importance of the adhesive force relative to the viscous force acting on a capsule, we introduce the dimensionless parameter Λ ) aN/γ˘ µRκ2, where γ˘ ) U0/Ly is the applied shear rate, and N ) 16πR2/(∆xs + ∆xc)2. Given that Ec is the Young’s modulus of the capsule’s shell, Φ ) aN/Echκ2 represents the ratio between the adhesive and elastic forces of the shell. Note that the capillary number for the capsule is given by Ca ) Φ/Λ. We systematically vary the stiffness of the shell and follow the capsule’s motion along a flat substrate that contains two stripes of width R, which are oriented R ) 45° relative to the flow direction. In the case of mechanical patterning, the capsulesubstrate adhesion is characterized by a ) 5.43 × 10-4 along the entire surface; however, the mechanical stiffness of the stripes is 100 times lower than the rest of the substrate. While the rolling capsules do not distort the stiffer regions, they can significantly deform the compliant strips (Figure 1). To facilitate our understanding of the mechanically patterned example, we also consider the conceptually simpler but equally interesting case of a rigid, but chemically patterned substrate where the strips display an adhesive strength of a ) 5.43 × 10-4, but the bulk of the surface exhibits a 5-fold decrease in the adhesive interaction, with a ) 1.09 × 10-4. The latter interaction strength is sufficiently strong to prevent the detachment of the capsules (by a lift force), but it exerts only a minor influence on the capsule’s shape. In contrast, adhesion within the patterned strips is sufficiently strong to cause a substantial deformation of (12) The parameters we chose for the capsules describe microcapsules formed by the layer-by-layer deposition method and certain biological cells; the parameters for the substrates describe polymeric surfaces. Further details on these choices and their relationship to experimental values are given in ref 10. We expect, however, generally similar behavior for elastic capsules and compliant microspheres generated by a variety of experimental approaches.
Letters
Figure 3. Trajectories of the center of mass motion for capsules with different compliancy on (a) chemically and (b) mechanically patterned substrates. The substrates encompass two consequent strips, which are oriented 45° relative to the flow direction. Starting from the same initial location, the capsules then move from left to right. As a result of the interaction with the patches, capsules of different stiffness gain different lateral displacements.
the capsule (Figure 2). (Note that we use the value of adhesive strength within the strip to calculate the dimensionless numbers.) Figure 3 shows the trajectories of the capsule’s center of mass motion on the chemically (Figure 3a) and mechanically (Figure 3b) patterned substrates. The distance between the centerlines of the two strips is 3R. We define ∆z2 as the total lateral displacement of the capsule, in the direction transverse to the fluid flow, after it has traversed both strips (see Figure 3). Although located at the same initial position, capsules with different Ec values attain remarkably different ∆z2, indicating that the capsule’s stiffness dictates its response to the patterned substrate. With chemical patterning, the softest capsule exhibits the largest lateral displacement (Figure 3a), while, with mechanical patterning, the rigid capsule attains the largest displacement ∆z2 (Figure 3b). The trajectories for different Ec nonetheless show some common trends, which we describe using the example of a rigid capsule on the chemically patterned surface. Driven by the fluid flow, the capsules initially roll on a homogeneous part of the substrate, and their trajectories are parallel to the x direction, corresponding to the a-b section in Figure 3a. As the capsules interact with the strip, they are “dragged” toward the centerline of this region, as indicated by the displacements in the negative z direction in section b-c. Upon reaching this centerline, the capsules’ lateral displacements change direction, as seen in section c-d. Now, the capsules tend to move within the strip, slowly approaching the strip boundary as they are propelled in the x direction by the imposed flow. As a result of traversing the entire strip, the capsules gain a net lateral displacement ∆z1. The capsules continue to move in a straightforward manner (section d-e) until they reach the second strip. Repeating the above dynamic behavior, the capsules attain an additional displacement of ∆z1, such that the total displacement over two strips yields ∆z2 ≈ 2∆z1. These results reveal that mechanical or chemical patterning
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Figure 4. Energy variation as a function of distance from the centerline of (a) adhesive and (b) soft patches for capsules with stiffness of Φ ) 2 (2) and Φ ) 1 (9), and for rigid capsules (b). The vertical lines mark the patches’ boundaries. From left to right, the snapshots illustrate capsules with Φ ) 2 at s ) 0, 0.7, and 2; colors indicate the strain as in Figures 1 and 2. The insets in panels a and b show the energy drop across the patch versus the capsule stiffness. Note that the adhesive strips and softer patches serve to lower the interaction energy between the capsule and the substrate. The drop in energy is affected by the width of the stripe. When the stripe is relatively narrow, the capsule can interact with both edges of the strip (see snapshots at s ) 0), reducing the overall energy drop. In the limit of very broad stripes (where the capsule essentially only feels one of the boundaries), the total energy change is defined by the difference in the energetic properties of the stripe and bulk material.
can be harnessed to direct the capsule’s lateral displacement on the substrate. Note that multiple stripes will enhance this effect. To rationalize these observations, we measure the energy W associated with the adhesive interaction between the capsule and substrate and the deformations due to this interaction. (The adhesive interaction energy is measured directly from the Morse potential acting between the substrate and the capsules; the energy associated with the elastic deformation is simply calculated from the strain within the spring network of our LSM.) To isolate the effects of adhesion, we calculate W in the absence of the imposed flow. In particular, we fix s, the distance between the capsule’s center of mass and the strip’s centerline, while allowing the capsule to equilibrate in the vertical (y) direction. For both types of patterning, Figure 4 shows the dependence of W on s for a single patch. Here, W is measured relative to the energy of a capsule on the homogeneous bulk of the substrate. In all cases, W has a minimum at s ) 0 and steadily increases with increasing s, meaning that the interaction energy depends on the change in the local properties of the substrate. The findings in Figures 3 and 4 imply that the strips introduce local energy minima into the system and thereby alter the energy landscape. The inherent energy minimization gives rise to a force on the capsules, Fw ∼ ∂W/∂s, which acts along the surface and
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Letters
Figure 5. Schematic of the capsule’s trajectory across a diagonal patch. (See explanation in the text.)
is directed toward the centerline. The superposition of Fw on the drag force due to the flow, Fd, controls the capsule’s dynamics as the capsule crosses the patterned substrate and causes the lateral displacement. (Recall that, in our calculations, Fw is rotated at 45° relative to the fluid flow and, therefore, to Fd.) Specifically, when the capsule approaches a strip, Fw tilts the capsule’s trajectory toward the strip’s centerline (see section b-c in Figure 3a). When the capsule reaches this centerline, Fw acts to localize the capsule within the more attractive or softer strip, altering the direction of the capsule’s motion (section c-d). Due to the asymmetry of this process, the capsule gains an overall displacement ∆z1 after passage over each of the strips, as seen in Figure 3. The value of ∆z1 is proportional to the magnitude of Fw, which, in turn, is related to ∆W, the amplitude of the energy variation over the strip. Although both types of surface patterning result in similar behavior for W (see Figure 4), the reasons for the energy variation with s are somewhat different. For the chemically patterned substrate, the decrease in energy at s ) 0 is primarily due to the following: (1) a stronger interaction within the more adhesive strip and (2) the increase in contact area A between the capsule and substrate due to a greater deformation of the capsule within the strip (see snapshots within Figure 4a). The value of A depends on the capsule’s elasticity; the softer shell experiences a larger change in the contact area and thus a larger variation in W when the capsule moves onto the adhesive strip (see inset in Figure 4a). Since Fw is proportional to the magnitude of the energy variation ∆W, the softer capsules gain a larger displacement ∆z2 as they roll along the substrate (Figure 3a). For the mechanically patterned, chemically uniform substrate, the largest contribution to ∆W comes from changes in contact area A; however, the energy of elastic deformations within the substrate and the capsule’s shell also contribute to ∆W. The change in A is primarily due to deformations that occur in the soft strip as it “envelops” the capsule (see snapshots within Figure 4b). The most dramatic change in A, and consequently W, occurs for a rigid capsule (see inset in Figure 4b). Thus, on the mechanically patterned substrate, Fw is lower for softer capsules, resulting in smaller values of ∆z2 (Figure 3b). The above arguments allow us to estimate the value of the capsule displacement ∆z1. A schematic of the capsule’s motion across a patterned substrate is shown in Figure 5. Before the capsule meets the strip (section a-b) and after it leaves the strip (section d-e), the capsule moves in the direction of the imposed flow with a velocity Ud ) (Ud, 0, 0), where Ud ∼ γ˘ Rav, and Rav is the average radius of a capsule deformed by its adhesion to the substrate. When the capsule is located within the strip, the capsule is also affected by the force due to the energy gradient Fw ∼ ∆W/∆s, where ∆s is now the effective range of the interaction between the capsule and the strip. The force Fw is balanced by the viscous interaction with the surrounding fluid. Thus, we estimate the velocity due to this force to be Uw ∼ Fw/32µRav, where the right-hand side of the expression is the
Figure 6. Capsule’s lateral displacement after moving over an (a) adhesive or (b) soft patch as a function of capsule stiffness for different shear rates. The symbols show the results of our simulations, while the lines represent the prediction of the scaling theory.
velocity of a sphere, which is near a wall, being transported by an external force in a viscous fluid.13 Note that the velocity has one direction Uw ) x2/2(Uw, 0, -Uw) when the capsule approaches the strip’s centerline (section b-c), and has the opposite direction, -Uw, after the capsule passes the centerline (section c-d). Knowing these capsule velocities and the geometry in Figure 5, we calculate ∆z1 ∼ 2x2Uw2∆s/(Ud2 - 2Uw2). Note that when Ud ) x2Uw, ∆z1f ∞; that is, the capsule cannot leave the strip. We, therefore, can estimate the minimal shear rate required for the capsule to cross the strip as γ˘ m ∼ (x2/32)Fw/µRav2. In Figure 6, we compare the predictions of the above model with the results of our simulations for different values of the shear rate γ˘ > γ˘ m. We set ∆s ) 1.6R (see Figure 4), and Rav is measured at s ) 0. For the case of chemical patterning (Figure 6a), we find surprisingly good agreement between the simple scaling theory and the simulations. Although the scaling theory correctly predicts the qualitative behavior on the mechanically patterned substrate, there is a noticeable difference from the simulation results for stiffer capsules (Figure 6b). This discrepancy arises because, in our scaling theory, we neglected the fact that a capsule moves slower on softer surfaces.9 Additionally, the stiff capsules are effectively submerged in the soft strip, thus reducing the surface area exposed to the external flow. This results in a reduction in Ud and an increase in Uw, thus enhancing the lateral displacement ∆z1 for such capsules. To facilitate the design of an efficient sorting device, it is worth noting that the sensitivity of the device can be controlled by simply varying the flow rate within the chamber.14,15 This point is illustrated in Figure 6 where we plot the dependence of ∆z1 on the capsule’s stiffness for different values of the shear rate γ˘ . Note that the data in Figure 6 can be used to extract the capsules’ stiffness Ech by measuring ∆z1, which is directly related (13) Goldman, A. J.; Cox, R. G.; Brenner, H. Chem. Eng. Sci. 1967, 22, 653.
Letters
to Φ. As shown in Figure 6, a decrease in shear rate γ˘ leads to an essential increase in ∆z1, indicating that reducing the flow rate enhances the sensitivity of the device. Thus, at relatively low γ˘ , even minute variations in capsule stiffness can be registered by the relatively large values of ∆z1. By comparing panels a and b of Figure 6, it is clear that the chemically patterned substrate produces a larger variation in ∆z1 (14) The sensitivity can also be adjusted via variations in the angle between the flow direction and the patterns. When the flow is perpendicular to the patterning, there is no effect on the capsule’s lateral motion; decreasing this angle will enhance the capsule’s lateral displacement. We note, however, that, below a critical value for the angle (that depends on flow rate), capsules with a given stiffness will be constrained to move along the patterned strip; their motion is not deflected by the imposed flow. (15) Note that the overall behavior can be affected by the roughness of the capsule’s shell and the substrate, as well as inhomogeneities on the surface of the cells; these effects are not considered in our model. Because of the roughness or inhomogeneity, capsules with identical stiffness may attain somewhat different lateral displacements, depending on the degree of the capsules’ imperfections.
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for softer capsules (Figure 6a), while, for the mechanical patterning, ∆z1 grows rapidly for Φ < 1 (Figure 6b). This indicates that the chemical patterning is optimal for separating softer capsules, whereas the mechanical patterning is more effective for segregating stiffer capsules. In either case, patterned substrates can be readily incorporated into microfluidic devices, and capsules of differing compliance can be continuously streamed over the surfaces, leading to a continuous sorting of these particles by their mechanical properties. Finally, we note that, since the energy variation ∆W depends on the local interaction strength between the capsule and substrate, our approach can also be used to separate microcapsules with identical mechanical properties, but different adhesiveness or binding energies. Acknowledgment. The authors gratefully acknowledge financial support from DOE (to A.A.) and ONR (to R.V.). LA062914Q
