Nonperturbative Evolution Equation for Quantum Gravity

A scale--dependent effective action for gravity is introduced and an exact nonperturbative evolution equation is derived which governs its renormalization group flow. It is invariant under general coordinate transformations and satisfies modified BRS Ward--Identities. The evolution equation is solved for a simple truncation of the space of actions. In 2+epsilon dimensions, nonperturbative corrections to the beta--function of Newton's constant are derived and its dependence on the cosmological constant is investigated. In 4 dimensions, Einstein gravity is found to be ``antiscreening'', i.e., Newton's constant increases at large distances.Comment: 35 pages, late

Supramolecular heterostructures formed by sequential epitaxial deposition of two-dimensional hydrogen-bonded arrays

Two-dimensional (2D) supramolecular arrays provide a route to the spatial control of the chemical functionality of a surface, but their deposition is in almost all cases limited to a monolayer termination. Here we investigated the sequential deposition of one 2D array on another to form a supramolecular heterostructure and realize the growth—normal to the underlying substrate—of distinct ordered layers, each of which is stabilized by in-plane hydrogen bonding. For heterostructures formed by depositing terephthalic acid or trimesic acid on cyanuric acid/melamine, we have determined, using atomic force microscopy under ambient conditions, a clear epitaxial arrangement despite the intrinsically distinct symmetries and/or lattice constants of each layer. Structures calculated using classical molecular dynamics are in excellent agreement with the orientation, registry and dimensions of the epitaxial layers. Calculations confirm that van der Waals interactions provide the dominant contribution to the adsorption energy and registry of the layers

Supramolecular networks stabilise and functionalise black phosphorus

The limited stability of the surface of black phosphorus (BP) under atmospheric conditions is a significant constraint on the exploitation of this layered material and its few layer analogue, phosphorene, as an optoelectronic material. Here we show that supramolecular networks stabilised by hydrogen bonding can be formed on BP, and that these monolayer-thick films can passivate the BP surface and inhibit oxidation under ambient conditions. The supramolecular layers are formed by solution deposition and we use atomic force microscopy to obtain images of the BP surface and hexagonal supramolecular networks of trimesic acid and melamine cyanurate (CA.M) under ambient conditions. The CA.M network is aligned with rows of phosphorus atoms and forms large domains which passivate the BP surface for more than a month, and also provides a stable supramolecular platform for the sequential deposition of 1,2,4,5-tetrakis(4-carboxyphenyl)benzene to form supramolecular heterostructures

Supramolecular nesting of cyclic polymers

Advances in template-directed synthesis make it possible to create artificial molecules with protein-like dimensions, directly from simple components. These synthetic macromolecules have a proclivity for self-organization that is reminiscent of biopolymers. Here, we report the synthesis of monodisperse cyclic porphyrin polymers, with diameters of up to 21 nm (750 C–C bonds). The ratio of the intrinsic viscosities for cyclic and linear topologies is 0.72, indicating that these polymers behave as almost ideal flexible chains in solution. When deposited on ​gold surfaces, the cyclic polymers display a new mode of two-dimensional supramolecular organization, combining encapsulation and nesting; one nanoring adopts a near-circular conformation, thus allowing a second nanoring to be captured within its perimeter, in a tightly folded conformation. Scanning tunnelling microscopy reveals that nesting occurs in combination with stacking when nanorings are deposited under vacuum, whereas when they are deposited directly from solution under ambient conditions there is stacking or nesting, but not a combination of both