Lamellar lattice

In connection to the data described earlier, one reasonable question arises Why does the stabilizing role of a nonaqueous solvent become significant for Cu salts and remain insignificant for Cu salts It was shown (Myagchenko et al. 1989) that Cu salts (and salts of Hg and Pd as well) exist in nonaqueous solutions as polynuclear compounds. Thus, CUCI2 forms lamellar lattices with chlorine chains as bridges between copper atoms. In contrast, CuCl does not form such chain structures. 

The entropy, S (298.15 K) = 18.585 cal K" mol", is obtained from the low temperature heat capacity data of Smith et al. (5), based on S (20 K) = 0.197 cal k" mol". This starting entropy was obtained by the authors from a T extrapolation of the data. Recently, McBride and Westrum (1 ) have shown that the T limiting law is Inappropriate for MoS and a T law is suggested for this and other compounds with lamellar lattices. They have also shown that the heat capacity data of Smith et al. for MoS... 

Fig. 8. /".p-phase diagram of DPPC bilayers in excess water and schematic drawing of the lamellar lattice constant and lipid packing in the bilayer plane of DPPC gel phases at 23 °C [44,85]. It is noteworthy that an additional crystalline gel phase (Lg) can be induced in the low-temperatue regime after prolonged cooling. 

The first two orders of the lamellar lattice are clearly resolved throughout the experiment, and in the low temperature L phase even the third order can be discovered above the noise. The peak intensities in the first order maxima are in the order of lO counts. This demonstrates that the time resolution of such an... 

So far, we have assumed that the crosslinks pin the smectic layers at a number of points but do not disturb the smectic density wave. However, a sufficient large density of crosslinks might lead to layer distortions that could destroy the quasi-long-range order of ID lamellar lattices [130, 131]. The crosslinks are randomly functionalized into the polymer backbone, and local density variations lead to quenched random disorder. This manifests itself as a mechanical random field that disturbs local layer positions and orientations. The effect of crosslinks on the smectic layer structure can be introduced via a corrugated potential that penalizes deviations of crosslinks from the local layer positions [4,132] ... 

Figure 7.7 Lamellar lattice constant of DEPC (l,2-dielaidoyl-glycero-3-phosphatidylcholine) in excess water after p-jumps inducing the L Lp transition (left) using different p-jump amplitudes and the Lp transition (top right). The series of diffraction patterns shown bottom right, taken at various time intervals after the p-jump, correspond to the same conditions as in the top-right figure. The emergence of the intermediate L, phase is clearly seen. Reproduced from Reference 33 with permission of the Royal Society of Chemistry.

The elliptical trace corresponds to the deformation of the lamellar lattice, and the disposition of the reflections along this trace reflects the orientation of the lamellae in this lattice [103]. One possibility is that as the lamellar spacing increases, the lateral distance between the adjacent lamellar columns decreases. Absence of any correlation between adjacent lamellar columns results in an eyebrow pattern (Fig. 2.16b). However, if the tilt... 

Micellar structure has been a subject of much discussion [104]. Early proposals for spherical [159] and lamellar [160] micelles may both have merit. A schematic of a spherical micelle and a unilamellar vesicle is shown in Fig. Xni-11. In addition to the most common spherical micelles, scattering and microscopy experiments have shown the existence of rodlike [161, 162], disklike [163], threadlike [132] and even quadmple-helix [164] structures. Lattice models (see Fig. XIII-12) by Leermakers and Scheutjens have confirmed and characterized the properties of spherical and membrane like micelles [165]. Similar analyses exist for micelles formed by diblock copolymers in a selective solvent [166]. Other shapes proposed include ellipsoidal [167] and a sphere-to-cylinder transition [168]. Fluorescence depolarization and NMR studies both point to a rather fluid micellar core consistent with the disorder implied by Fig. Xm-12. 

Figure B3.6.4. Illustration of tliree structured phases in a mixture of amphiphile and water, (a) Lamellar phase the hydrophilic heads shield the hydrophobic tails from the water by fonning a bilayer. The amphiphilic heads of different bilayers face each other and are separated by a thin water layer, (b) Hexagonal phase tlie amphiphiles assemble into a rod-like structure where the tails are shielded in the interior from the water and the heads are on the outside. The rods arrange on a hexagonal lattice, (c) Cubic phase amphiphilic micelles with a hydrophobic centre order on a BCC lattice.

Lattice models have been studied in mean field approximation, by transfer matrix methods and Monte Carlo simulations. Much interest has focused on the occurrence of a microemulsion. Its location in the phase diagram between the oil-rich and the water-rich phases, its structure and its wetting properties have been explored [76]. Lattice models reproduce the reduction of the surface tension upon adsorption of the amphiphiles and the progression of phase equilibria upon increasmg the amphiphile concentration. Spatially periodic (lamellar) phases are also describable by lattice models. Flowever, the structure of the lattice can interfere with the properties of the periodic structures. 

This method has been devised as an effective numerical teclmique of computational fluid dynamics. The basic variables are the time-dependent probability distributions f x, f) of a velocity class a on a lattice site x. This probability distribution is then updated in discrete time steps using a detenninistic local rule. A carefiil choice of the lattice and the set of velocity vectors minimizes the effects of lattice anisotropy. This scheme has recently been applied to study the fomiation of lamellar phases in amphiphilic systems [92, 93]. 

Lattice models for bulk mixtures have mostly been designed to describe features which are characteristic of systems with low amphiphile content. In particular, models for ternary oil/water/amphiphile systems are challenged to reproduce the reduction of the interfacial tension between water and oil in the presence of amphiphiles, and the existence of a structured disordered phase (a microemulsion) which coexists with an oil-rich and a water-rich phase. We recall that a structured phase is one in which correlation functions show oscillating behavior. Ordered lamellar phases have also been studied, but they are much more influenced by lattice artefacts here than in the case of the chain models. 

FIG. 13 Phase diagram of a vector lattice model for a balanced ternary amphiphilic system in the temperature vs surfactant concentration plane. W -I- O denotes a region of coexistence between oil- and water-rich phases, D a disordered phase, Lj an ordered phase which consists of alternating oil, amphiphile, water, and again amphi-phile sheets, and L/r an incommensurate lamellar phase (not present in mean field calculations). The data points are based on simulations at various system sizes on an fee lattice. (From Matsen and Sullivan [182]. Copyright 1994 APS.)... 

Langevin simulations of time-dependent Ginzburg-Landau models have also been performed to study other dynamical aspects of amphiphilic systems [223,224]. An attractive alternative approach is that of the Lattice-Boltzmann models, which take proper account of the hydrodynamics of the system. They have been used recently to study quenches from a disordered phase in a lamellar phase [225,226]. 

Typical runs consist of 100 000 up to 300 000 MC moves per lattice site. Far from the phase transition in the lamellar phase, the typical equilibration run takes 10 000 Monte Carlo steps per site (MCS). In the vicinity of the phase transitions the equilibration takes up to 200 000 MCS. For the rough estimate of the equihbration time one can monitor internal energy as well as the Euler characteristic. The equilibration time for the energy and Euler characteristic are roughly the same. For go = /o = 0 it takes 10 000 MCS to obtain the equilibrium configuration in which one finds the lamellar phase without passages and consequently the Euler characteristic is zero. For go = —3.15 and/o = 0 (close to the phase transition) it takes more than 50 000 MCS for the equihbration and here the Euler characteristic fluctuates around its mean value of —48. 

The first section involves a general description of the mechanics and geometry of indentation with regard to prevailing mechanisms. The experimental details of the hardness measurement are outlined. The tendency of polymers to creep under the indenter during hardness measurement is commented. Hardness predicitions of model polymer lattices are discussed. The deformation mechanism of lamellar structures are discussed in the light of current models of plastic deformation. Calculations... 

Recent developments have allowed atomic force microscopic (AFM) studies to follow the course of spherulite development and the internal lamellar structures as the spherulite evolves [206-209]. The major steps in spherulite formation were followed by AFM for poly(bisphenol) A octane ether [210,211] and more recently, as seen in the example of Figure 12 for a propylene 1-hexene copolymer [212] with 20 mol% comonomer. Accommodation of significant content of 1-hexene in the lattice allows formation and propagation of sheaf-like lamellar structure in this copolymer. The onset of sheave formation is clearly discerned in the micrographs of Figure 12 after crystallization for 10 h. Branching and development of the sheave are shown at later times. The direct observation of sheave and spherulitic formation by AFM supports the major features that have been deduced from transmission electron and optical microscopy. The fibrous internal spherulite structure could be directly observed by AFM. 

Ruland and Smarsly [84] study silica/organic nanocomposite films and elucidate their lamellar nanostructure. Figure 8.47 demonstrates the model fit and the components of the model. The parameters hi and az (inside H ) account for deviations from the ideal two-phase system. Asr is the absorption factor for the experiment carried out in SRSAXS geometry. In the raw data an upturn at. s o is clearly visible. This is no structural feature. Instead, the absorption factor is changing from full to partial illumination of the sample. For materials with much stronger lattice distortions one would mainly observe the Porod law, instead - and observe a sharp bend - which are no structural feature, either. 

---