Crumpled membranes

Figure 14.7 (a) Crumpled membrane module due to excessive pressure drop or water hammer (b) cracked membrane module. 

While the Flory approximation works well for some systems, this is not always the case. Two such examples are Sierpihski gaskets ind=2 and two-dimensional tethered membranes in = 3. For tethered membranes, the Flory-level theory predicts that df = 5/2 or d = D = 2 and d= 5. This state is commonly referred to as a crumpled membrane since the relaxed membrane should be isotropic. This expectation that the membrane would crumple was supported by renormalization group calculations and early MC simulations on small systems. However a number of more detailed and simulations found that the membranes... 

Break up the crumpled PVDF membrane into a few pieces using a disposable needle... 

The way in which salts such as calcium chloride and calcium formate operate is not fully understood, but it is clear that the mechanism involves an acceleration of the C S and C S hydration. It has been proposed [23] that the initial products of cement hydration form a sort of membrane which acts as a restraint to the diffusion process which in turn leads to the dormancy period . It seems likely that the chloride ion, by virtue of its small size and high mobility, is able more easily to penetrate the pores of the restraining layer allowing the diffusion process to proceed more rapidly. The resultant tobermorite gel has a higher lime-silica ratio and a more open, accessible structure, based on a crumpled foil morphology rather than the usual spicular. The considerable reaction with,... 

Flexible, solid membranes, are also of interest. However, they are experimentally much less prevalent and are somewhat more complicated to treat since in addition to the membrane shape one must include the effects of shear. Their curvature energy is discussed in the problems at the end of this chapter. Another type of system that has received much theoretical attention is that of a tethered membrane which may describe polymerized, but not crystalline sheets. While a single fluid membrane that is unconstrained by walls or other membranes is strongly affected by thermal fluctuations ( crumpled ), solid membranes, particularly if self-avoidance of the membrane is included, tend to be more weakly affected by fluctuations and are hence flattef . 

From these analyses, it is clear that if the attractive interaction is finite ranged, then the membrane without self-avoiding interaction has the possibility to exhibit a continuous crumpling transition from the flat phase to the compact phase and there may exist a critical crumpled state at the transition point. However, in order to discuss the nature of the transition we must compare the corresponding free energies between the flat phase and the compact phase. It is a difficult task and we only make some reasoning based on the numerical studies of ref. [4] hereafter. 

In this section, we review some of the recent progress in simulating polymeric fractals, with particular emphasis on tethered membranes made of linear polymer segments connected together to form a two-dimensional surface. After a brief review of the theory, we present results from a munber of groups which show that two-dimensional tethered membranes remain flat and do not crumple. We then consider the effect of changing the solvent quality by adding attractive interactions between nonbonded monomers. While there is clear evidence for a collapsed phase at low T, the nature of the crossover from flat to compact state remains unclear. 

One of the original motivations for studying large membranes made of linear chains was to investigate whether reducing the local bending rigidity would cause the membrane to crumple. Abraham showed very conclusively in his study of open membranes that the order parameter, deflned as the ratio of square of the membrane size, vanishes only as the... 

While it is now commonly accepted that self-avoiding membranes without attractive interactions remain flat and do not crumple, the flat phase still exhibits some interesting properties. For a membrane containing N monomers, the transverse displacements have an amplitude where... 

The morphology and shape of a sheet play crudal roles in its spedfic functions. A platelet or tethered membrane [33, 40-44] exhibits unique conformational characteristics as it wrinkles and crumples under various physical constraints, such as quality of solvent and temperature, while entropy dissipates from its open boundaries. As pointed out above, one can tether partides - the smallest units in a coarsegrained description - to form both sheets and other hierarchical units such as chains. 

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