A molecular model for the line tension of lipid membranes

The line tension of a symmetric, lipid bilayer in its liquid-crystalline state is calculated on the basis of a molecular lipid model. The lipid model extends the opposing forces model by an expression for the conformational free energy of the hydrocarbon chains. We consider a membrane edge that consists of a perturbed bilayer covered by a section of a cylinder-like micelle. The structural rearrangement of the lipids implies an excess free energy which we minimize with respect to the cross-sectional shape of the membrane edge, including both the micellar and the bilayer region. The line tension is derived as a function of molecular lipid properties, like the lipid chain length or the head group interaction strength. We also relate it to the spontaneous curvature of the lipid layer. We find the line tension to become smaller for lipid layers that tend to curve more towards the hydrophobic core. Our predictions for the line tension and their relation to experimentally derived values are discussed. Received 2 January 2000     

Undulation instability in two-dimensional foams of magnetic fluid

Two-dimensional magnetic fluid foams are cellular structures whose framework is made of magnetic fluid. The features of these equilibrated patterns are driven by a control parameter: the amplitude of the applied magnetic field. When the latter is rapidly increased, an instability occurs: the walls between cells undulate. Such an instability has also been observed in other 2D cellular structures, which exist for instance in Langmuir monolayers or in magnetic garnets thin films. In this paper we give a theoretical analysis of this instability, the issues of which are shown to be well confirmed by experiments and numerical simulations. Received: 13 October 1997 / Received in final form: 28 January 1998 / Accepted: 9 March 1998     

Tilt model of inverted amphiphilic mesophases

We present an alternative model of structure and energetics of the inverted amphiphilic mesophases. The previous studies of the inverted hexagonal, H_II, and inverted micellar cubic, Q_II, phases considered the amphiphilic monolayers to be homogeneously bent. In contrast, we assume a unit cell of an inverted mesophase to consist of flat fragments of monolayer. Hence, the unit cells of the H_II and Q_II phases are represented by a hexagonal rod and a polyhedron, respectively. Our model is motivated by Turner and Gruner's X-ray diffraction reconstruction of structure of the H_II phase. The only deformation of the amphiphilic monolayers we consider is tilt of the hydrocarbon chains with respect to the monolayer surface, determined by the packing constraints imposed in the mesophases. Applying our recent model for the elastic energy of tilt in liquid membranes [#!ref23!#], we show that: i) tilt accounts in a natural way for the frustration energy of mesophases resulting from filling by the hydrocarbon chains the corners of the unit cells, ii) the energy of tilt variation along the membrane surface is analogous to the bending energy. We compute the energetics of the H_II, Q_IIsc and Q_IIfcc phases and obtain a hypothetical phase diagram in terms of the elastic constants of monolayers. Moreover, we calculate the structural dimensions of the mesophases. We verify the model showing that the obtained phase diagram describes the recent data for the glycolipids/water systems; the predicted dimensions of the Q_II phase are in accord with the measured values; the model treats quantitatively the structural features observed for the H_II phase. Received: 9 February 1998 / Revised: 4 June 1998 / Accepted: 3 July 1998     

Stiffening of fluid membranes and entropy loss of membrane closure: Two effects of thermal undulations

The problem of membrane softening by thermal undulations is revisited. In contrast to general belief, fluid membranes are predicted to be stiffened, not softened, by their undulations. Equal values of the effective bending rigidity are calculated from the interplay of local mean curvature modes (hats) on the basically flat membrane and from the coupling of spherical harmonic modes with spherical curvature. In addition, a conjecture is made on the entropy of membrane closure. It relies on a similarity of membrane closure to periodic boundary conditions. Received: 10 June 1997 / Revised: 7 October 1997 / Accepted: 2 December 1997     

The segregation of poly(styrene-b-isoprene) diblock copolymers to the surface of a polystyrene melt: the effect of the ratio of block lengths

The effect of the ratio of block lengths on the interfacial partitioning of poly(styrene-block-1,4 isoprene) diblock copolymers from their mixtures with polystyrene homopolymer melt is investigated utilizing a series of copolymers with almost constant molecular weight but different compositions. The concentration profile of the copolymer is measured directly using the nuclear reaction analysis technique; a segregation of the diblock is found at both the air/polymer surface, due to the lower surface energy of polyisoprene, and at the substrate/polymer interface. No significant effect of the block length ratio on the free-surface excess was observed. The block molecular weights have apparently led to dangling chain conformations in the non-overlapping mushroom and in the overlapping mushroom regimes whereas the brush regime was not accessible; no indications of a real border between the two former regimes was found. Received: 20 July 1998 / Received in final form and Accepted: 11 September 1998     

Shear viscosity of polymer and surfactant monolayers

The surface shear viscosity of monolayers formed at the surface of water by adsorbed polyethyl- eneoxyde and by stearic acid is measured as a function of the surface pressure of the monolayer using a new surface viscometer. The principle of the viscometer is the measurement of the drag force on a circular disk undergoing a uniform translation at the water surface: a hydrodynamic model based on the lubrication approximation allows a calculation of the surface viscosities from the absolute measurement of the drag forces. Received: 26 August 1999     

Elastic energy of tilt and bending of fluid membranes

Tilt of hydrocarbon chains of lipid molecules with respect to membrane plane is commonly considered to characterize the internal structure of a membrane in the crystalline state. However, membranes in the liquid state can also exhibit tilt resulting from packing constraints imposed on the lipid molecules in diverse biologically relevant structures such as intermediates of membrane fusion, pores in lipid bilayers and others. We analyze the energetics of tilt in liquid membranes and its coupling with membrane bending. We consider three contributions to the elastic energy: constant tilt, variation of tilt along the membrane surface and membrane bending. The major assumption of the model is that the core of a liquid membrane has the common properties of an elastic continuum. We show that the variation of tilt and membrane bending are additive and that their energy contributions are determined by the same elastic coefficient: the Helfrich bending modulus, the modulus of Gaussian curvature and the spontaneous curvature known from previous studies of pure bending. The energy of a combined deformation of bending and varying tilt is determined by an effective tensor accounting for the two factors. In contrast, the deformation of constant tilt does not couple with bending and its contribution to the elastic energy is determined by an independent elastic constant. While accurate determination of this constant requires additional measurements, we estimate its value using a simplified approach. We discuss the relationships between the obtained elastic Hamiltonian of a membrane and the previous models of membrane elasticity. Received 10 February 2000 and Received in final form 19 June 2000     

Microscopic measurement of the linear compressibilities of two-dimensional fatty acid mesophases

The linear compressibility of two-dimensional fatty acid mesophases has been determined by grazing incidence X-ray diffraction. The unit cell parameters of the , , , S and phases of behenic acid and of the phase of myristic acid were determined as a function of surface pressure and temperature. Surface pressure versus molecular area isotherms were reconstructed from these measurements, and the linear compressibility (relative distortion along a given direction for a two-dimensional isotropic applied stress) was determined both in the sample plane and in a plane normal to the aliphatic chain director (transverse plane). The linear compressibilities range over two orders of magnitude from 0.1 to 10 m/N and are distributed depending on their magnitude in 4 different sets which we are able to associate with different molecular mechanisms. The largest compressibilities (10 m/N) are observed in the tilted phases. They are apparently independent on the chain length and could be related to the reorganization of the headgroup hydrogen-bounded network, whose role should be revalued. Intermediate compressibilities are observed in phases with quasi long-range order (directions normal to the molecular tilt in the or phases, S phase, and could be related to the ordering of these phases. The lowest compressibilities are observed in the solid untilted phase and for one direction of the S and phases. They are similar to the compressibility of crystalline polymers and correspond to the interactions between methyl groups in the crystal. Finally, negative compressibilities are observed in the transverse plane for the and phases and can be traced to subtle reorganizations upon untilting. Received: 29 July 1997 / Revised: 14 October 1997 / Accepted: 23 October 1997     

Secondary buckling patterns of a thin plate under in-plane compression

We study the nonlinear deformations of a long rectangular elastic plate clamped along its edges and submitted to in-plane biaxial compression. Using the F?ppl-von Kármán equations, we predict various secondary buckling modes according to the applied longitudinal and transverse compressions. A model experiment is carried out in a thin polycarbonate film, and the observed buckling patterns are found in good agreement with theory. Pattern selection in the delamination of compressed thin films is discussed in the light of these results. Received 22 January 2002     

Optical transmittance spectroscopy of concentric-shell fullerenes layers produced by carbon ion implantation

Concentric-shell fullerenes, also called carbon onions, produced by carbon ion implantation into silver thin films, and subsequently deposited on a silica substrate, were studied by optical transmission spectroscopy in the wavelength range 0.2 - 1.2 μm. In this interval, the strongest absorption is due to the π-plasmon of sp²-like carbon. The position of the plasmon absorption band clearly evolved from 265 nm at low fluence to 230 nm at high implantation fluences. A simulation of the optical spectra based on dielectric models of the concentric-shell fullerenes layer allowed us to identify the first peak as due to disordered graphite and the latter to the carbon onions. The concentration of residual graphite and the filling fraction of the carbon onions produced at high fluences could be estimated by fitting the optical spectra with computed transmittance curves. Received 13 July 2000     

Ascending air bubbles in protein solutions

We report measurements of the ascending velocity of air bubbles in protein (bovine serum albumin) solutions. We show that, because of the protein molecules adsorbed on their surface, the terminal ascending velocity of bubbles is strongly reduced compared to the terminal velocity in pure water: protein- covered bubbles behave hydrodynamically as solid spheres. From the evolution of the ascending velocity with time, we can derive the amount of protein needed to immobilize the bubble interface which is 0.5 mg m^-2, i.e. only one fifth of the amount adsorbed at equilibrium in the range of used bulk concentrations. Received: 6 March 1998 / Revised and accepted: 6 May 1998     

Effect of thermal undulations on the bending elasticity and spontaneous curvature of fluid membranes

We amplify previous arguments why mean curvature should be used as measure of integration in calculating the effective bending rigidity of fluid membranes subjected to a weak background curvature. The stiffening of the membrane by its fluctuations, recently derived for spherical shapes, is recovered for cylindrical curvature. Employing curvilinear coordinates, we then discuss stiffening for arbitrary shapes, confirm that the elastic modulus of Gaussian curvature is not renormalized in the presence of fluctuations, and show for the first time that any spontaneous curvature also remains unchanged. Received 19 April 1999 and in Received in final form 7 January 2000     

Thermoelastic behaviour of polyvinylacetate monolayers at the air-water interface: Evidences for liquid-solid phase transition

The thermoelastic behaviour of polyvinylacetate monolayers spread on an aqueous subphase has been studied using rheological data previously published (Monroy et al., Phys. Rev. E 58, 7629 (1998)). The results show fluid-like viscoelastic behaviour well above a transition temperature , while at lower temperatures a soft solid-like behaviour emerges. The correlation between thermodynamic and elastic properties below can be described in terms of scaling laws. Received 12 January 1999 and Received in final form 11 June 1999     

Molecular resolution of an organic monolayer by dynamic AFM

We report molecularly resolved dynamic AFM-measurements of a monolayer of perylene-3,4,9,10-tetracarboxylic-3,4,9,10-dianhydride (PTCDA) epitaxially grown on Ag(110). Preparation and experiments were performed under UHV conditions. The frequency-modulation AFM technique was applied. Received: 26 February 1998 / Revised: 13 May 1998 / Accepted: 8 June 1998     

Falling ball viscosimetry of giant vesicle membranes: Finite-size effects

We study the general problem of the friction felt by a spherical solid particle which moves parallel to the membrane of a spherical vesicle. Experiments are carried out with SOPC vesicles at room temperature, with different particle and vesicle sizes. Experimental data show considerable finite-size effects whenever the particle is not very small compared to the vesicle. These effects are found consistent with the hydrodynamical theory of the vesicle-particle problem. This agreement allows for a “robust” determination of membrane viscosity, independently of particle and vesicle sizes. Received 4 January 1999 and Received in final form 11 May 1999     

Growth instabilities of vapor deposited films: atomic size versus deflection effect

Two previously suggested, physically distinct mechanisms for a growth instability of vapor deposited films, the finite atomic size effect and the particle deflection effect due to interatomic attraction, are reconsidered, further analyzed, and compared. We substantiate why the instability caused by interatomic attraction must be considered as the truly underlying instability mechanism. We demonstrate that aspects of the structure zone model of Movchan and Demchishin can also be consistently explained using the growth instability induced by particle deflection instead of the instability arising from the atomic size effect. Most significantly we show that, for vapor deposited amorphous Zr₆₅Al_7.5Cu_27.5-films, the growth instability due to the atomic size effect cannot be present. Received 12 December 2001 Published online 6 June 2002     

Kinetics of shape equilibration for two dimensional islands

We study the relaxation to equilibrium of two dimensional islands containing up to 20 000 atoms by Kinetic Monte Carlo simulations. We find that the commonly assumed relaxation mechanism - curvature-driven relaxation via atom diffusion - cannot explain the results obtained at low temperatures, where the island edges consist in large facets. Specifically, our simulations show that the exponent characterizing the dependence of the equilibration time on the island size is different at high and low temperatures, in contradiction with the above cited assumptions. Instead, we propose that - at low temperatures - the relaxation is limited by the nucleation of new atomic rows on the large facets: this allows us to explain both the activation energy and the island size dependence of the equilibration time. Received 7 December 1998 and Received in final form 18 March 1999     

X-ray scattering by nano-particles within granular thin films, investigation by grazing angle X-ray reflectometry

It is shown here that the observation of the phenomenon of like small angle scattering of X-rays in very thin heterogeneous films, can be made comparatively easily by using grazing angle reflectometry of X-rays. The feasibility was achieved with co-sputtered thin films of approximately 600 ? thickness, made up by crystalline platinum clusters embedded in an amorphous alumina matrix. The experimental reflectivity profiles are simulated by the intensity superposition of two components: (i) the specular part caused by the usual interference phenomenon between the partial waves reflected from the air-film and film-substrate interfaces, and (ii) the like-small angle scattering part due to diffraction by platinum clusters. It is found that the shape of such clusters is spherical characterized by mean values of diameter and inter-cluster distance of the order 29 ? and 45 ? respectively with standard deviations and of the order of 3 ?. Such an observation of both the interference and diffraction phenomena indicates that the thin granular film exhibits both its continuous and heterogeneous aspects together. Received: 19 February 1998 / Accepted: 25 August 1998     

3D Large scale Marangoni convection in liquid films

We study large scale surface deformations of a liquid film unstable due to the Marangoni effect caused by external heating on a smooth and solid substrate. The work is based on the thin film equation which can be derived from the basic hydrodynamic equations. To prevent rupture, a repelling disjoining pressure is included which accounts for the stabilization of a thin precursor film and so prevents the occurrence of completely dry regions. Linear stability analysis, nonlinear stationary solutions, as well as three-dimensional time dependent numerical solutions for horizontal and inclined substrates reveal a rich scenario of possible structures for several realistic fluid parameters.     

Elastic interaction between defects in thin and 2D films

Elastic interactions between defects is investigated at the surface of thin layers, a question on which we have given a brief account [P. Peyla et al. Phys. Rev. Lett. 82, 787 (1999)]. Two isotropic defects do not interact in an unlimited medium, regardless of the spatial dimension, a result which can be shown on the basis of the Gauss theorem in electrostatics. Within isotropic elasticity theory, defects interact only (i) if they are, for example, at a surface (or at least if they feel a boundary), or if their action on the material is anisotropic (e.g. they create a non central force distribution, though the material elasticity is isotropic). It is known that two identical isotropic defects on the surface of a semi-infinite material repel each other. The repulsion law behaves as 1/r ³(r = defects separation). We first revisit the Lau-Kohn theory and extend it to anisotropic defects. Anisotropy is found to lead to attraction. We show that in thin films defects may either attract or repel each other depending on the local geometric force distribution caused by the defect. It is shown that the force distribution (or more precisely the forces configuration symmetry) fixes the exponent in the power law 1/r ⁿ (e.g. for a four-fold symmetry n = 4). We discuss the implication of this behaviour in various situations. We treat the interactions in terms of the symmetries associated with the defect. We argue that if the defects are isotropic, then their effective interaction in an unlimited 2D (or a thin film) medium arises from the induced interaction, which behaves as 1/r ⁴ for any defect symmetry. We shall also comment on the contribution to the interaction which arises from flexion of thin films. Received 7 October 2002 Published online 4 June 2003 RID="a" ID="a"e-mail: chaouqi.misbah@ujf-grenoble.fr