Self-Assembly of Mesoscale Objects into Ordered Two-Dimensional Arrays

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Science  11 Apr 1997:
Vol. 276, Issue 5310, pp. 233-235
DOI: 10.1126/science.276.5310.233


Regular arrays of topologically complex, millimeter-scale objects were prepared by self-assembly, with the shapes of the assembling objects and the wettability of their surfaces determining the structure of the arrays. The system was composed of solid objects floating at the interface between perfluorodecalin and water and interacting by lateral capillary forces; patterning of the wettability of the surfaces of the objects directs these forces. Self-assembly results from minimization of the interfacial free energy of the liquid-liquid interface. Calculations suggest that this strategy for self-assembly can be applied to objects on a micrometer scale.

This report describes the directed self-assembly of small objects (between 1 and 10 mm in length) into regular, two-dimensional arrays. It extends ideas emerging from molecular self-assembly to much larger objects and describes an approach for the fabrication of complex systems that has potential for application in microelectronics, optics, microelectromechanical systems, and displays (1). These self-assembled systems have analogies to bubble rafts (2, 3) and related aggregates formed by spheres on surfaces (4, 5); they differ in that the use of nonspherical objects with designed shapes and wettabilities makes it possible to generate aggregates with complex structures.

The individual objects were made of polydimethylsiloxane (PDMS), a hydrophobic polymer with a surface free energy (γSV) equal to 22 to 24 erg cm−2 (6-8). First, the PDMS was cured in a mold of the desired shape. Designated surfaces were then made hydrophilic by oxidation with an O2 plasma (6, 9); surfaces that were to be hydrophobic were either covered with tape before the oxidation or generated by cutting of the PDMS after oxidation to expose a fresh hydrophobic surface. For each of the objects in Fig. 1, the lower face [the face in contact with the perfluorodecalin (C10F18)] was hydrophobic, and the sides were either hydrophobic or hydrophilic according to the design summarized in the insets. The upper face was usually made hydrophilic to prevent any C10F18from settling on top of the objects and causing them to sink into the C10F18-H2O interface. The C10F18 wetted the hydrophobic sides and formed menisci; the hydrophilic sides barely perturbed the C10F18-H2O interface (10,11). The system was placed on a rotary shaker that oscillated gently in the plane of the interface. The frequency of rotation was adjusted for each system to allow the in-plane movement of the liquid to bring the objects close to one another and to break up misformed aggregates; a typical frequency was 1 to 2 Hz. When two hydrophobic sides of PDMS came within a critical distance of one another (∼5 mm), they moved into contact. This movement was driven by the minimization of the interfacial free energy of the system, caused by eliminating the curved menisci at the hydrophobic surfaces. A thin layer of C10F18 remained between the objects, even at their closest contact (12). Self-assembly was completed after ∼5 to 30 min.

Figure 1

Crystalline aggregates generated by the self-assembly of (A) crosses, (B) hexagons in an open network, and (C) hexagons close-packed. Panel (A) shows an extended two-dimensional square array formed from crosses having hydrophobic ends; the two crosses with arrows pointed toward them were completely oxidized; these pieces were rejected by the array. They clearly show the shape of the individual pieces. Hatched faces in the insets indicate hydrophobic surfaces and white faces indicate hydrophilic surfaces. The top faces are clear and hydrophilic.

A variety of self-assembled arrays can be made by controlling the hydrophobicity of the sides of the objects (Fig. 1A). The contrast between Fig. 1B and Fig. 1C demonstrates the level of control that we can achieve: In Fig. 1B, alternate sides of the hexagons are hydrophobic and the objects form an open network; in Fig. 1C, all of the sides of the hexagons are hydrophobic and the objects form a close-packed array.

A shape-selective lock-and-key geometry allows the formation of aggregates from more than one component (Fig. 2, A through C). The objects in Fig. 2A have three favorable choices for contact (Fig. 2C). The amplitude of oscillation of the system during equilibration was set to be large enough to break apart pairs interacting head-to-head and tail-to-tail but to leave together those interacting more strongly head-to-tail. Head-to-tail contact is favored energetically by two complementary factors: it maximizes the area of hydrophobic surface in close proximity; and it is kinetically stable to dissociation caused by stirring, because the oscillating motion at the interface—a motion that seems to influence the objects primarily by shear—has minimal influence on pairs of objects once assembled into a head-to-tail configuration.

Figure 2

Lock-and-key self-assembly. Lines are formed from one type of component (A) and from two different types of components (B). (C) The three possibilities for favorable assembly of two objects in (A). Although in the tail-to-tail arrangement in (C) no hydrophobic surfaces are in contact, this interaction was observed to be important because the menisci are brought close enough together to interact. Dark faces in the insets in (A) and (B) indicate hydrophobic sides and white faces indicate hydrophilic sides. The top faces are clear and hydrophilic.

A third method for self-assembly uses the area of hydrophobic side surfaces, and thus the strength of the attractive capillary force, to direct the self-assembly of different objects (Fig. 3, A through D). A mixture of two types of PDMS objects with the same square bases, but heights that differed by a factor of 5, were agitated at the interface between C10F18 and water. The order of the attractive forces in the system was tall-tall > tall-short > short-short. The degree of agitation was set to allow the tall objects to form an array; when agitation was stopped, the short objects coagulated around this array in a disordered state.

Figure 3

Self-assembly based on different heights. The two components are identical except for their heights, and their sides are hydrophobic. The light gray objects in (A) and (B) have a height of 2.5 mm and the dark gray objects have a height of 0.5 mm. In (A) the objects have just begun to assemble. After ∼20 min of agitation, the objects in (B) have segregated into two regions: a crystalline central array of the taller objects, surrounded by a disordered collection of the smaller ones. (C) A schematic view of the interaction of the objects of different heights. (D) Side view of the three different interactions when two pieces of PDMS with different heights are assembled. The strength of interaction increases from right to left.

The above experiments were carried out with objects having dimensions of 1 to 10 mm. We investigated the lower limits to the size of the objects that could be assembled by lateral capillary forces at the C10F18-H2O interface by calculating the change in interfacial free energy as two perpendicular surfaces moved from infinite separation to some finite separation, d. We calculated the height h (in meters) of the C10F18-H2O interface between the two objects using the linearized Laplace equation (13) (Fig.4A)Embedded Image (1)where γ (in joules per square meter) is the interfacial free energy, Δρ (in kilograms per cubic meter) is the difference in density between the two fluids, the zero for h is set at the C10F18-H2O interface far from the objects, g (in meters per second per second) is the acceleration due to gravity, and ΔP0 (in pascals) is the change in pressure across the interface at x = 0. If we assign a value of h(0) = 0, then the value of ΔP0 does not enter into the solution of Eq.1. Using the boundary conditions h(r) = t, where t (in meters) is the thickness of the object and (∂h/∂x)x=0 = 0, we find that the solution of Eq. 1 isEmbedded Image (2)where we have made the replacement xc = (γ/Δρg)1/2. For infinite d the capillary surface is given by a simple exponential decay with h(x) ∝ e(−x/xc ).

Figure 4

(A) The coordinate system for Eqs. 1 and2. The objects have a height of t and a width ofw = 5t, and their proximate surfaces are separated by d. (B) The logarithm of the change in interfacial free energy—divided by thermal energy,kT—in bringing two surfaces from infinite separation to a finite separation, d, is plotted for heights fromt = 1 mm to 100 nm.

To estimate the change in interfacial energy as a function of distance, we calculated the difference in the arc length Δℓ (in meters), defined by h(x) for two surfaces separated by d and two surfaces separated by an infinite distance; the change in arc length was then multiplied by the width of the object w (in meters), and the interfacial free energy to yield the change in interfacial free energy (14). As a model system, we assigned a length to the perpendicular surface equal to five times the height. This model gave a change in interfacial free energy, ΔW, of ΔW = 5Δℓγt. From the change in interfacial free energy for heights from t = 1 mm to 100 nm (Fig.4B), we conclude that the energetics for self-assembly are favorable for objects with t as small as 100 nm. For the two-dimensional self-assembly of spheres, the radius at which ΔW/kT ∼ 1 (where kT is the thermal energy) has been calculated to be on the order of 1 to 10 μm (15, 16). Self-assembly driven by capillary forces between conformal surfaces should therefore make possible the assembly of much smaller objects than would be possible with spheres; the ability to control the shapes and interfacial properties of these objects makes it possible to design the geometries of the resulting arrays.

Four factors contribute to the success of this strategy for the directed self-assembly of small objects. First, the aggregates are energetically more stable than the individual dissociated objects or disordered aggregates. Second, formation of the aggregates is reversible when the system is agitated: formation and dissociation of the aggregates compete. The aggregates are therefore able to reach the energetically most stable form. Third, the hydrophobic sides are attracted to one another over large distances (about two to three times the dimension of the height), leading to relatively rapid assembly. Fourth, even when two hydrophobic sides are in close proximity, they can move laterally from side to side, lubricated by the intervening film of C10F18, and can thus maximize the amount of hydrophobic area in contact.


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