Self-Assembly of Mesoscale Objects in ordered 2D Arrays - The ...

fluence the objects primarily by shear—hasminimal influence on pairs of objects onceassembled into a head-to-tail configuration.A third method for self-assembly usesthe area of hydrophobic side surfaces, andthus the strength of the attractive capillaryforce, to direct the self-assembly of differentobjects (Fig. 3, A through D). A mixture oftwo types of PDMS objects with the samesquare bases, but heights that differed by afactor of 5, were agitated at the interfacebetween C 10 F 18 and water. The order of theattractive forces in the system was talltall tall-short short-short. The degreeof agitation was set to allow the tall objectsto form an array; when agitation wasFig. 2. Lock-and-key self-assembly. Lines areformed from one type of component (A) and fromtwo different types of components (B). (C) Thethree possibilities for favorable assembly of twoobjects in (A). Although in the tail-to-tail arrangementin (C) no hydrophobic surfaces are in contact,this interaction was observed to be importantbecause the menisci are brought close enoughtogether to interact. Dark faces in the insets in (A) and (B) indicate hydrophobic sides and white facesindicate hydrophilic sides. The top faces are clear and hydrophilic.stopped, the short objects coagulatedaround this array in a disordered state.The above experiments were carried outwith objects having dimensions of 1 to 10mm. We investigated the lower limits to thesize of the objects that could be assembledby lateral capillary forces at the C 10 F 18 -H 2 Ointerface by calculating the change in interfacialfree energy as two perpendicularsurfaces moved from infinite separation tosome finite separation, d. We calculated theheight h (in meters) of the C 10 F 18 -H 2 Ointerface between the two objects using thelinearized Laplace equation (13) (Fig. 4A) 2 hx 2 1 gh P 0 (1)where (in joules per square meter) is theinterfacial free energy, (in kilograms percubic meter) is the difference in densitybetween the two fluids, the zero for h is setat the C 10 F 18 -H 2 O interface far from theobjects, g (in meters per second per second)is the acceleration due to gravity, and P 0(in pascals) is the change in pressure acrossthe interface at x 0. If we assign a valueof h(0) 0, then the value of P 0 does notenter into the solution of Eq. 1. Using theboundary conditions h(r) t, where t (inmeters) is the thickness of the object and(h/x) x0 0, we find that the solution ofEq.1is2hxt ex/xc d/2xc e x/xc)1e d/xc e d/2xc (2) eFig. 3. Self-assembly based on different heights. Thetwo components are identical except for their heights,and their sides are hydrophobic. The light gray objectsin (A) and (B) have a height of 2.5 mm and thedark gray objects have a height of 0.5 mm. In (A) theobjects have just begun to assemble. After 20 minof agitation, the objects in (B) have segregated intotwo regions: a crystalline central array of the tallerobjects, surrounded by a disordered collection of thesmaller ones. (C) A schematic view of the interaction of the objects of different heights. (D) Side view ofthe three different interactions when two pieces of PDMS with different heights are assembled. Thestrength of interaction increases from right to left.Fig. 4. (A) The coordinate system for Eqs. 1 and 2.The objects have a height of t and a width of w 5t , and their proximate surfaces are separated byd. (B) The logarithm of the change in interfacialfree energy—divided by thermal energy, kT—inbringing two surfaces from infinite separation to afinite separation, d, is plotted for heights from t 1 mm to 100 nm.234SCIENCE VOL. 276 11 APRIL 1997 http://www.sciencemag.org