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The subtle interactions between fluid-filled polymeric shells and compliant surfaces lie at the heart of a number of technologically important and fundamentally intriguing problems. Consider, for instance, microcapsules that consist of a fluid enclosed in a thin, polymeric casing. Such microencapsulated fluids are becoming increasingly important in the pharmaceutical, cosmetics, and food industries. For example, microcapsules containing anticancer drugs can be tailored to target tumor cells or lesions.1 In such applications, the adhesion of microcapsules onto surfaces can be essential to their functionality.2 However, the dynamic interactions between the enclosed fluid and the polymer casing can affect the binding of the microcapsule to the desired surface3 and thus, must be taken into account in designing efficacious delivery systems.

The motion of biological cells along surfaces provides another example of complex fluid-surface interactions involving polymers. Within the fluidic interior of the cell, actin is polymerized to form a rigid polymer. This rigid polymer then effectively “prods” the outer “shell” or cellular membrane to form a protrusion (filopodia) that extends along a compliant substrate. (While this is a simplified scenario, it nonetheless captures salient features.) Thus, just as in the microcapsule example, the cell's internal fluid-surface interactions affect its interaction with the substrate.

In addition to understanding the cells' in vivo behavior, to perform various biological assays and tissue engineering studies, it is vital to comprehend and control the cells ex vivo dynamic behavior.4 Polymeric materials can play a critical role in this arena, where there is a need for “smart” surfaces that can effectively modulate the motion of cells and thereby allow them to be readily sorted, isolated, or encapsulated.

To design efficacious microcapsules, obtain a more complete understanding of cellular motility and fabricate smart substrates for manipulating cells, one crucially needs theoretical/computational models that capture not only the fluid-membrane interactions within the microcapsules or cells, but also the interactions between these fluid-filled shells and the targeted interfaces. The significant challenge is capturing the relevant dynamic interactions in a consistent and computationally efficient manner.

Before suggesting possible pathways for addressing this issue, we first note that the interactions between flowing fluids and the surrounding, compliant walls are commonly referred to as “fluid-structure” interactions. Conventional numerical solutions to fluid-structure interaction problems generally involve the coupling of a finite element method for the structural analysis with a finite difference, finite volume, or finite element method for the computational fluid dynamics. While the solid and fluid subsystems can be solved simultaneously,5 typically an approach is adopted that involves separate computer codes for the solid and fluid systems that are solved iteratively to a solution that satisfies both.6 The iterative process is repeated until consistent results are obtained, which satisfy the constraints of both the fluid and the solid domain.

Recently, however, lattice-based simulation techniques have emerged, which are computationally more efficient than their continuum-based counterparts. Unlike conventional numerical schemes, which involve a direct discretization of the continuum equations, these lattice models simulate the underlying processes that give rise to the appropriate continuum behavior. In particular, the lattice Boltzmann model (LBM) incorporates the mesoscopic physics of fluid “particles” propagating and colliding on a simple lattice, such that the averaged, macroscopic properties of the system obey the desired Navier-Stokes equations.7 In a similar fashion, the lattice spring model (LSM) is adopted from atomistic models of solid-state and molecular physics,8 and involves a network of interconnected “springs,” which describe the interactions between neighboring units. The large-scale behavior of the resultant system can be mapped on to continuum elasticity theory.9 The LBM and LSM are, therefore, both mesoscopic models, whose local rules are guided by microscopic phenomena, but whose emergent behavior captures the macroscopic properties of the system.

In a recent study,3 we took advantage of these mesoscale approaches to formulate a new technique for modeling fluid-solid interactions. In particular, we coupled the LBM and LSM to simulate the three-dimensional behavior of a single-phase fluid that is enclosed in a homogeneous, elastic spherical shell. This approach allows for a dynamic interaction between the elastic walls and the enclosed fluid. In other words, dynamically and interactively, the moving walls exert a force on the fluid, and in turn, the confined fluid reacts back on the walls. We then utilized this model to investigate the binding and interactions between these microcapsules and various hard, flat surfaces.

This hybrid LBM/LSM approach provides one methodology for addressing a number of the challenges alluded to herein, and we spotlight this technique as a means of focusing the ensuing narrative. However, other modeling methods10, 11 could also be harnessed to examine these important issues. Let us start by considering a liquid-filled elastic shell that is surrounded by a flowing Newtonian fluid and lies in contact with a surface. One issue that needs to be addressed is introducing the compliancy or responsiveness of the surface; this is vital for modeling, for example, the interactions between skin and microcapsules in health care formulations or cells and the endothelial layer. In addition, it is necessary to determine how variations in the external flow field affect the vesicle-substrate interactions.

The first of the above features can be captured through the LSM component of the hybrid approach. Specifically, the dynamic LSM consists of a network of harmonic springs that connect regularly spaced mass points. By varying the spring constants, one can alter the relative stiffness or compliance of the substrate. In this manner, one can determine how the mechanical characteristics of this interface affect the adhesion and motion of the vesicle on the surface (see Fig. 1).

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Figure 1. Fluid-filled shell rolling on a soft substrate. The shell spring constants are roughly 70 times greater than those for the substrate. The shell is driven by an imposed shear and attracted to the surface by a Morse potential. The arrows mark the flow velocities; the size of the arrows indicates the magnitude of these velocities. Thin lines on the shell and substrate represent non-diagonal lattice springs. The colors denote the magnitude of the strain, ε. The values in the color bar correspond to a range of ε = 0.5 to ε = −0.5 for the substrate and ε = 0.05 to ε = −0.05 for the shell.

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The second of the above issues can be addressed through the LBM. An imposed shear can cause the vesicle to “roll” along the compliant surface; other flows can be harnessed to streamline the motion of the capsules. Furthermore, by modifying the relative relaxation times in the LBM, the viscosities of the host and encapsulated can be altered, and consequently, one can analyze how such differences in the fluids' features affect the interactions between the vesicles and the surfaces.

To completely characterize the utility of microcapsules, which act as delivery systems, one must also model how interactions with the substrate can cause the outer shell to burst and thereby release its vital contents. In the dynamic LSM, springs can be selectively removed, when, for example, the energy associated with the extension of the springs becomes sufficiently high.12 In this manner, one could potentially simulate the rupture of the outer shell when the microcapsule binds to and is distorted by contact with the surface (see the significant strain variations within the shell in Fig. 1). We note that the LBM can be used to simulate multicomponent fluid. Consequently, the hybrid approach can capture not only the interactions between the enclosed and host fluids, but also the dynamic interactions between the released fluid and the targeted surface.

An important challenge is modeling the viscoelasticity of the microcapsule's polymeric shell or of an underlying polymeric surface. One means of addressing this issue is to introduce a dissipative unit between the springs that connect the lattice sites. In particular, a Kelvin unit, which is composed of a spring in parallel with a dashpot, can be connected in series with each spring in the system.13 This approach has been successfully implemented in the LSM to capture the viscoelastic response of polymeric materials.14 Another important issue has to do with modeling the non-Newtonian nature of the encapsulated fluid. A number of authors have proposed schemes for describing the viscoelasticity of the fluid through the LBM scheme.15–18 Consequently, these approaches could potentially be harnessed to model the rheological behavior of encapsulated polymeric fluids.

Both of the above issues are relevant to modeling the complex fluid-structure interactions involving biological cells (i.e. capturing the flow of the jelly-like cytoplasm and the viscoelasticity of the cell membrane). In this context, we now return to the intriguing question of interrelating the polymerization of actin into a rigid chain within the cell, the interaction of this rigid chain with the cellular membrane and the subsequent motion of the cell along a compliant surface. While the LBM/LSM approach could provide a framework for constructing such an integrated model, further theoretical developments will be required to simulate this complex system. For example, an LB scheme has recently been developed to model reactive mixtures that undergo the reversible chemical reaction A + B [LEFT RIGHT ARROW] C.19 However, the free energy functional for the system must be modified to reflect the fact that C is a growing chain. One then also has to specify the rigidity of the polymer in the free energy expression. Another challenge is to capture the locality of the interaction between the polymerized actin and the cellular membrane; namely, that the membrane protrudes locally in response to pressure from the growing, rigid polymer chain. Next, one would hope to capture how this filopodia grips an underlying biological surface, and how in turn, the soft layer responds to the pressure of this “foot”. Although this is a clearly coarse-grained view of the complex phenomena involved in cell motility, the studies could nonetheless provide insight into the forces and surface tensions that play a critical role in the process.

Up to this point, we have focused our discussion on the interactions between a single vesicle and the substrate. However, it is also intriguing to consider the interactions among multiple vesicles, which are also situated on a compliant interface. Through these studies, one could establish design criteria for driving synthetic cells to aggregate in response to variations in the surface or surrounding solution (see Fig. 2). In particular, one can envision designing an entire circuit of vesicles that are driven to move, merge, and exchange their contents. In doing so, these directed microcapsules could act as microreactors that carry out specific reactions at specified locations, that is, they form a micron-scale assembly plant.

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Figure 2. Rolling of two fluid-filled shells on a rigid substrate. The shear-driven shells are attracted to each other and to the green part of the substrate via a Morse potential. The arrows show the flow velocities. Red dots are drawn to help illustrate the rotation of the shells. (a) Initial position of the shells; (b) Shell 2 reaches the end of the attractive (green) region and stops; (c) Shell 1 approaches and adheres to shell 2; and (d) The agglomerate detaches from the end of the attractive region due to the shear flow and travels downstream.

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We introduce another important dimension of the problem when we consider the flow of vesicles in capillaries or narrow channels. Now, these capsules experience the effects of confinement due to the presence of additional surfaces. Such studies are vital for understanding how variations in the mechanical properties or geometry of the channels can lead to abnormalities in biological systems or in the processing of emulsions and suspensions. Here, it is essential to capture not only the interactions between a deformable capsule and the compliant walls, but also the effects of both the host and encapsulated fluids.

In summary, this an opportune time for probing the dynamic interactions between vesicles (synthetic or biological) and compliant surfaces, since advances in mesoscale modeling make it possible to perform efficient computational studies. The findings from these studies can permit the design of microcapsules for targeted delivery of vital drugs, and can help enhance our fundamental understanding of the complex biomechanics involved in cellular motility. In addition, the studies can provide guidelines for creating smart polymeric substrates that can actively regulate the in vitro motion of cells, thereby facilitating the further development of biosensors or scaffolds for tissue engineering.4

Acknowledgements

  1. Top of page
  2. Acknowledgements
  3. REFERENCES AND NOTES

The author gratefully acknowledges helpful discussions with Alexander Alexeev, Rolf Verberg, David Jasnow and Gavin A. Buxton. The results shown in Figures 1 and 2 were carried out in collaboration with Alexander Alexeev and Rolf Verberg.

REFERENCES AND NOTES

  1. Top of page
  2. Acknowledgements
  3. REFERENCES AND NOTES