Here we describe a new and simple method for preparing alkyl monolayers on silicon, which consists of mechanically scribing oxide-coated silicon while it is wet with 1-alkenes or 1-alkynes (neat or in inert solvents) under ambient conditions. X-ray photoelectron spectroscopy, time-of-flight secondary ion mass spectrometry, wetting data, and stability tests suggest covalent bonding of unsaturated species to exposed silicon surfaces. Enclosures (hydrophobic corrals) made by scribing silicon that is wet with unsaturated hydrophobic species hold droplets of water and liquids with substantially lower surface tensions. Wetting tests suggest that 1-alkynes make better hydrophobic corrals than 1-alkenes, and theoretical results suggest it should be more difficult for alkyl chains of chemisorbed 1-alkenes to pack than those of 1-alkynes. Underivatized interior regions of hydrophobic corrals are functionalized with polyelectrolyte multilayers. Theoretical energies for water and methanol droplets (gravitational and surface) in hydrophobic corrals are calculated, and a model of failure of liquid droplets in hydrophobic corrals is presented.

Abstract. The first author proved that the harmonic convolution of a normalized right half-plane mapping with either another normalized right halfplane mapping or a normalized vertical strip mapping is convex in the direction of the real axis. provided that it is locally univalent. In this paper, we prove that in general the assumption of local univalency cannot be omitted. However, we are able to show that in some cases these harmonic convolutions are locally univalent. Using this we obtain interesting examples of univalent harmonic maps one of which is a map onto the plane with two parallel slits.

Abstract. Ruscheweyh and Sheil-Small proved that convexity is preserved under the convolution of univalent analytic mappings in K. However, when we consider the convolution of univalent harmonic convex mappings in K O H , this property does not hold. In fact, such convolutions may not be univalent. We establish some results concerning the convolution of univalent harmonic convex mappings provided that it is locally univalent. In particular, we show that the convolution of a right half-plane mapping in K O H with either another right halfplane mapping or a vertical strip mapping in K O H is convex in the direction of the real axis. Further, we give a condition under which the convolution of a vertical strip mapping in K O H with itself will be convex in the direction of the real axis.

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