Curvature elastic moduli

Lekkerkerker, H. N. W. (1990). The electric contribution to the curvature elastic moduli of charged fluid interfaces, Physica A, 167, 384—394. 

Next, we consider the flexural properties of surfactant adsorption monolayers, which are important for the formation of small droplets, micelles, and vesicles in the fluid dispersions. The contributions of various interactions (van der Waals, electrostatic, steric) into the interfacial bending moment and the curvature elastic moduli are described. The effect of interfacial bending on the interactions between deformable emulsion droplets is discussed. 

Below, we first introduce the most general mechanical description of the surface moments (torques) exerted on the boundary between two fluid phases. Then, we consider the thermodynamics of a curved interface (membrane) in terms of the work of flexural deformation. Next, we specify the bending rheology by means of the model of Helfiich [202]. Finally, we review the available expressions for the contributions of the electrostatic, steric, and van der Waals interactions to the interfacial bending moment and curvature elastic moduli. These expressions relate the interfacial flexural properties to the properties of the adsorbed surfactant molecules. 

E. Physical Importance of the Bending Moment and the Curvature Elastic Moduli... 

Expressions for the electrostatic components of the curvature elastic moduli, and k, in terms of the nonlinear double-layer theory can be found in Ref. 227. Contributions of point dipoles, diffuse and Stem parts of the electic double layer in Bq, are evaluated in Ref. 210. Alternatively, one can relate Bq, k, and to the surface Volta potential, AV, which is a directly measurable parameter [213] ... 

The mechanical properties of surfactant adsorption monolayers are characterized not only by the interfacial tension but also by the interfacial bending moment, which is proportional to the so-called spontaneous curvature of the interface. In addition, the variation of the interfacial bending moment with curvature is characterized by the curvature elastic moduli. These interfacial flexural properties are determined mostly by the interactions between the head groups and tails of the adsorbed surfactant molecules. In their own turn, the interfacial flexural properties influence phenomena and processes such as the formation of microemulsions, critical emulsions, holes in foam and emulsion Aims, fluctuation capillary waves, flocculation in emulsions, and so on see Sec. IV. 

In order to express the condition for spontaneous vesiculation in terms of curvature elastic moduli, we start from the usual formula for the bending energy per unit area of fluid bilayer... 

Splay ( bending or curvature ) defined by a splay constant K, or by a curvature elastic modulus = K h [76],... 

Figure 4 The modified stalk mechanism of membrane fusion and inverted phase formation, (a) planar lamellar (La) phase bilayers (b) the stalk intermediate the stalk is cylindrically-symmetrical about the dashed vertical axis (c) the TMC (trans monolayer contact) or hemifusion structure the TMC can rupture to form a fusion pore, referred to as interlamellar attachment, ILA (d) (e) If ILAs accumulate in large numbers, they can rearrange to form Qn phases, (f) For systems close to the La/H phase boundary, TMCs can also aggregate to form H precursors and assemble Into H domains. The balance between Qn and H phase formation Is dictated by the value of the Gaussian curvature elastic modulus of the bIlayer (reproduced from (25) with permission of the Biophysical Society) The stalk in (b) is structural unit of the rhombohedral phase (b ) electron density distribution for the stalk fragment of the rhombohedral phase, along with a cartoon of a stalk with two lipid monolayers merged to form a hourglass structure (reproduced from (26) with permission of the Biophysical Society).

An application of the optical microscopy to the detennination of the curvature elastic-modulus of biological and model membranes. Journal of Physics, 48 (5). 855-867. 

Bivas, P. Hanusse, P. Bothorel, J. Lalanne, and O. Aguerre-Chariol,/. Phys. (Paris), 48,855 (1987). An Application of the Optical Microscopy to the Determination of the Curvature Elastic Modulus of Biological and Model Membranes. 

These fascinating bicontinuous or sponge phases have attracted considerable theoretical interest. Percolation theory [112] is an important component of such models as it can be used to describe conductivity and other physical properties of microemulsions. Topological analysis [113] and geometric models [114] are useful, as are thermodynamic analyses [115-118] balancing curvature elasticity and entropy. Similar elastic modulus considerations enter into models of the properties and stability of droplet phases [119-121] and phase behavior of microemulsions in general [97, 122]. 

This measures the curvature about an axis perpendicular to the dispersion plane, i.e. the cylindrical curvature, and it may be necessary to rotate the wafer through 90° to get the orthogonal component. This may be related to absolute stress in the wafer with knowledge of the wafer thickness, diameter and elastic modulus. The most accurate method is to measure a number of points on a wafer and use a linear regression formula for the average curvature. 

A Models to describe microparticles with a core/shell structure. Diametrical compression has been used to measure the mechanical response of many biological materials. A particular application has been cells, which may be considered to have a core/shell structure. However, until recently testing did not fully integrate experimental results and appropriate numerical models. Initial attempts to extract elastic modulus data from compression testing were based on measuring the contact area between the surface and the cell, the applied force and the principal radii of curvature at the point of contact (Cole, 1932 Hiramoto, 1963). From this it was possible to obtain elastic modulus and surface tension data. The major difficulty with this method was obtaining accurate measurements of the contact area. 

Bianko and Marmur [99] have developed a new technique for the measurement of Gibbs elasticity of foam films. In order to exclude the effect of the mass transfer of the surfactant, the stretching of an isolated soap bubble is used. The surface tension needed for the calculation of the elasticity modulus is determined by the pressure in the bubble and the radius of curvature. The modulus obtained are considerably lower than those derived by the technique of Prins et al. [95]. 

Fig.5 shows the calculated curvature and temperature evolution for an FGM deposit with thickness of about 180 im, which is consistent with the experimental results shown in Fig.4 except for the transient oscillations. Fig.6 (a) shows the calculated stress distributions in 2-layer and FGM deposits. The gradual stress variation in the FGM can be observed. In Fig.6 (b) effects of model parameters such as the substrate temperature and elastic modulus of YPSZ on the stress distribution in 2-layer deposits are demonstrated. As the substrate temperature is raised from 600 to 825K, the tensile stress in the NiCrAlY layer is significantly reduced. If a value of elastic modulus of 190GPa of a dense bulk material was used, the compressive residual stress in the YPSZ is excessively overestimated. This example clearly demonstrates the importance of using realistic values for modeling thermal and mechanical behavior of sprayed deposits. 

Explain why the curvature elastic constants vanish in the limit of vanishing shear modulus. Include in your explanation a consideration of why one expects k and in a small molecule liquid i.e., with no springlike chain molecules such as those considered in the text) to be very small. 

As far as we are aware, the only system for which the volume fraction of the droplets, the average radius (R32), and the tension of the macroscopic oil/water interface have been measured is aWinsor II system composed of SDS, pentanol, cyclohexane, and 0.2 M NaCl with equal volumes of water and oil phases. This system was studied in Ref 55. The bending elastic modulus of this system was measured by ellipsometry [50], The results of Sec. V imply that when / and z are fixed, there are still two unknown parameters the Gaussian bending elastic modulus and the preferred curvature. Therefore, we choose to test the theory on consistency, that is, we fix z and / and fit the (R32, 4>) with Eqs. (55)-(58)—we choose... 

We have already mentioned in the Introduction that microemulsions containing long-chain amphiphiles can be described by interfacial models. These models are based on the curvature elasticity of the amphiphilic monolayer and thus contain as material parameters the bending rigidity and the saddle-splay modulus. These parameters have to be calculated from a more microscopic model. A somewhat similar problem occurs in the... 

The exceptional interfacial properties of amphiphilic molecules are due to the opposing affinities of their polar and hydrophobic parts with respect to water. A remarkable consequence of these properties is that we can produce transparent and macroscopically homogeneous mixtures of oil and water, the so-called microemulsions. Such thermodynamically stable mixtures are not, of course, homogeneous on microscopic scales. Here we can observe water and oil microdomains separated by amphiphilic molecular films. Stability and structure of microemulsions are determined by properties of this amphiphilic film in particular, by the elasticity of its curvature. The relevant parameters are its elastic modulus K and its spontaneous curvature cq. 

Fig. 5.23, the mean curvature must be as close as possible to cq < 0, and indeed, it is symmetrically negative. When co < 0, the saddle surface is therefore preferred in comparison with the plane surface. These intuitive geometrical considerations can be re-expressed in terms of the Gaussian curvature modulus K of the bilayer. The curvature elasticity of the bilayer (Kbiiayer, biiayer and Co, bilayer = 0) Can bc simply calculated in terms of the curvature elasticity of the monolayers (i monoiayer) -f monoiayer and Cq). The problem is as simple as for a bimetallic strip. We obtain... 

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