Polymers and colloids change the 'spontaneous' curvature of flexible membranes such as lipid bilayers. Two general cases will be discussed: (i) The effect of single polymers or colloids which are anchored or adsorbed to the membranes and exert bending moments on the adjacent membrane segments; and (ii) the effect of unbalanced tensions in the two membrane/water interfaces which arise, e.g., from polymer brushes and from non-anchored polymers or colloids.
Bundles of strings which interact via short-ranged pair potentials are studied in two dimensions. The corresponding transfer matrix problem is solved analytically for arbitrary string number N by Bethe ansatz methods. Bundles consisting of N identical strings exhibit a unique unbinding transition. If the string bundle interacts with a hard wall, the bundle may unbind from the wall via a unique transition or a sequence of N successive transitions. In all cases, the critical exponents are independent of N and the density profile of the strings exhibits a scaling form that approaches a mean-field profile in the limit of large N .PACS numbers: 68.10, 64.70, 82.70-y In the context of condensed matter physics, strings are essentially 1-dimensional objects which are (i) directed in the sense that their tangent vectors point, on average, into a certain direction, and (ii) [5]. Two different ensembles of strings have to be distinguished: (i) systems with a fixed density of strings and (ii) systems with a fixed number N of strings, which are the topic of this letter. If the strings have attractive interactions, they may at low temperatures be bound together to a bundle. Such bundles of strings have been studied by numerical diagonalization of the transfer matrix [6,7], in a local density functional theory [8], by mapping onto a quantum spin chain [9], in a heuristic scaling picture [10], and by field-theoretic renormalization group methods [11].
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