Membrane-mediated interactions of rod-like inclusions

Inclusions embedded in lipid membranes undergo a mediated force, due to the tendency of the membrane to relax its excess of elastic energy. In this paper we determine the exact shape of a two-dimensional vesicle hosting two different inclusions, and we analyse how the inclusion conformation influences the mediated interaction. We find non-trivial equilibrium configurations for the inclusions along the hosting membrane, and we derive the complete phase diagram of the mediated interaction. In particular, we find a non-vanishing mediated force even when the distance between the inclusions is much greater than their size. Our model can be applied to describe the mediated interactions of parallel, elongated inclusions embedded in three-dimensional membranes. Received 22 October 2001 and Received in final form 8 March 2002     

Primitive interactions between inclusions on a fluid membrane: The role of thermal fluctuations

Consider a fluid membrane decorated by moving hard or soft inclusions. The aim of this work is a quantitative study of the influence of thermal fluctuations on the three-dimensional primitive forces between these inclusions. Integrating over all membrane fluctuations, we obtain a general form giving the modified primitive interactions upon the transverse distance. The established formalism enables us to obtain the modified expression of some standard interaction potentials. In particular, for power-like potentials, we found a modified expression featuring the Whittaker function. The present formalism may be extended to other primitive interaction potentials. Finally, the main conclusion is that, decorated fluid membranes may be regarded as effective two-dimensional colloidal solutions where inclusions interact via the computed effective interactions.     

Effective Hamiltonian approach to doubly degenerate electronic states

Effective vibronic Hamiltonian models are built for E ⊗e Jahn-Teller systems and analytical solutions are obtained through Lie algebraic methods. Although approximate, we show that these models allow in particular to recover the possible ground state crossover when quadratic couplings are present. The equivalence of E ⊗e and G' ⊗e vibronic systems in cubic symmetry is precisely established through a particular realization of the electronic operators for an orbital quadruplet. We show how this equivalence is broken by a rovibronic interaction which, for a G' ⊗e system, still gives an exactly solvable model.     

Quantitative Pointwise Estimate of the Solution of the Linearized Boltzmann Equation

We study the quantitative pointwise behavior of the solutions of the linearized Boltzmann equation for hard potentials, Maxwellian molecules and soft potentials, with Grad’s angular cutoff assumption. More precisely, for solutions inside the finite Mach number region (time like region), we obtain the pointwise fluid structure for hard potentials and Maxwellian molecules, and optimal time decay in the fluid part and sub-exponential time decay in the non-fluid part for soft potentials. For solutions outside the finite Mach number region (space like region), we obtain sub-exponential decay in the space variable. The singular wave estimate, regularization estimate and refined weighted energy estimate play important roles in this paper. Our results extend the classical results of Liu and Yu (Commun Pure Appl Math 57:1543–1608, 2004), (Bull Inst Math Acad Sin 1:1–78, 2006), (Bull Inst Math Acad Sin 6:151–243, 2011) and Lee et al. (Commun Math Phys 269:17–37, 2007) to hard and soft potentials by imposing suitable exponential velocity weight on the initial condition.     

Why is nacre strong? II. Remaining mechanical weakness for cracks propagating along the sheets

In our previous paper (Eur. Phys. J. E 4, 121 (2001)) we proposed a coarse-grained elastic energy for nacre, or stratified structure of hard and soft layers found in certain seashells . We then analyzed a crack running perpendicular to the layers and suggested one possible reason for the enhanced toughness of this substance. In the present paper, we consider a crack running parallel to the layers. We propose a new term added to the previous elastic energy, which is associated with the bending of layers. We show that there are two regimes for the parallel-fracture solution of this elastic energy; near the fracture tip the deformation field is governed by a parabolic differential equation while the field away from the tip follows the usual elliptic equation. Analytical results show that the fracture tip is lenticular, as suggested in a paper on a smectic liquid crystal (P.G. de Gennes, Europhys. Lett. 13, 709 (1990)). On the contrary, away from the tip, the stress and deformation distribution recover the usual singular behaviors ( and 1/, respectively, where x is the distance from the tip). This indicates there is no enhancement in toughness in the case of parallel fracture. Received 16 November 2001     

N-body study of anisotropic membrane inclusions: Membrane mediated interactions and ordered aggregation

We study the collective behavior of inclusions inducing local anisotropic curvatures in a flexible fluid membrane. The N-body interaction energy for general anisotropic inclusions is calculated explicitly, including multi-body interactions. Long-range attractive interactions between inclusions are found to be sufficiently strong to induce aggregation. Monte Carlo simulations show a transition from compact clusters to aggregation on lines or circles. These results might be relevant to proteins in biological membranes or colloidal particles bound to surfactant membranes. Received 30 July 1999 and Received in final form 8 September 1999     

Attractive forces between anisotropic inclusions in the membrane of a vesicle

The fluctuation-induced interaction between two rod-like, rigid inclusions in a fluid vesicle is studied by means of canonical ensemble Monte-Carlo simulations. The vesicle membrane is represented by a triangulated network of hard spheres. Five rigidly connected hard spheres form rod-like inclusions that can leap between sites of the triangular network. Their effective interaction potential is computed as a function of mutual distance and angle of the inclusions. On account of the hard-core potential among these, the nature of the potential is purely entropic. Special precaution is taken to reduce lattice artifacts and the influence of finite-size effects due to the spherical geometry. Our results show that the effective potential is attractive and short-range compared with the rod length L. Its well depth is of the order of , where is the bending modulus. Received 5 February 1999 and Received in final form 14 May 1999     

Two direct methods to calculate fluctuation forces between rigid objects embedded in fluid membranes

The fluctuation-induced attractive interaction of rigid flat objects embedded in a fluid membrane is calculated for a pair of parallel strips and a pair of equal circular disks. Assuming flat boundary conditions, we derive the interaction from the entropy of the suppressed boundary angle fluctuation modes. Each mode entropy is computed in two ways: from the boundary angles themselves and from the mean-curvature mode functions. A formula for the entropy loss of suppressing one or more mean-curvature modes is developed and applied. For the pair of disks we recover the result of Goulian et al. and Golestanian et al. in a direct manner, avoiding any mappings by Hubbard-Stratonovitch transformations. The mode-by-mode agreement of the two computed entropies in both systems confirms an earlier claim that mean curvature is the natural measure of integration for fluid membranes. Received 15 December 2000     

Three unequal masses on a ring and soft triangular billiards

The dynamics of three soft interacting particles on a ring is shown to correspond to the motion of one particle inside a soft triangular billiard. The dynamics inside the soft billiard depends only on the masses ratio between particles and softness ratio of the particles interaction. The transition from soft to hard interactions can be appropriately explored using potentials for which the corresponding equations of motion are well defined in the hard wall limit. Numerical examples are shown for the soft Toda-like interaction and the error function.