Lamellar thickness, fluctuation

Fig. 3.16a-c. a A series of fluctuations which are allowed in Frank s model, that is, attachments and detachments are both allowed but the chain is not permitted to fold until it has reached the lamellar thickness, b and c show a series of events which may also be expected to occur... 

A further paper [167] explains the lamellar thickness selection in the row model. The minimum thickness lmin is derived from the similation and found to be consistent with equilibrium results. The thickness deviation 81 = l — lmin is approximately constant with /. It is established that the model fulfills the criteria of a kinetic theory Firstly, a driving force term (proportional to 81) and a barrier term (proportional to /) are indentified. Secondly, the competition between the two terms leads to a maximum in growth rate (see Fig. 2.4) which is located at the average thickness l obtained by simulation. Further, the role of fluctuations becomes apparent when the dependence on the interaction energy e is investigated. Whereas downwards (i.e. decreasing l) fluctuations are approximately independent... 

Sect. 4.1). In polymers, lamellar thickness is kinetically determined by the restricted upward fluctuation in stem length on the one hand and, on the other hand, by the high detachment rate of shorter stems with l -> /mjn = 2ae/(A(p) (downward fluctuation). This problem has been studied analytically [62,74] and by simulation [71,75]. 

This potential force occurs in microstructured fluids like microemulsions, in cubic phases, in vesicle suspensions and in lamellar phases, anywhere where an elastic or fluid boundary exists. Real spontaneous fluctuations in curvature exist, and in liposomes they can be visualised in video-enhtuiced microscopy [59]. Such membrane fluctuations have been invoked as a mechanism to account for the existence of oil- or water-swollen lamellar phases. Depending on the natural mean curvature of the monolayers boimding an oil region - set by a mixture of surfactant and alcohol at zero -these swollen periodic phases can have oil regions up to 5000A thick With large fluctuations the monolayers are supposed to be stabilised by steric hindrance. Such fluctuations and consequent steric hindrance play some role in these systems and in a complete theory of microemulsion formation. 

Porod predicted that, for an ideal lamellar system of two phases (Fig. 19.5) in which neither fluctuations of density within phases nor interfacial thickness of finite wide are present, the intensity of dispersion diminishes proportionally to the reciprocal of the fourth power of q, which is mathematically expressed as... 

We can imagine a cholesteric as a smck of nematic quasi-layers of molecular thickness a with the director slightly turned by ( ) from one layer to the next one. In fact it is Oseen model [18]. Such a structure is, to some extent, similar to lamellar phase. Indeed, the quasi-nematic layers behave like smectic layers in formation of defects, in flow experiments, etc. Then, according to the Landau-Peierls theorem, the fluctuations of molecular positions in the direction of the helical axis blur the one-dimensional, long-range, positional (smectic A phase like) helical order but in reality the corresponding scale for this effect is astronomic. 

Fig. 7.1. Evolution of orientation fluctuations that occur when an amorphous polymer evolves to a lamellar crystal. This process can be divided into two distinct processes (I) formation of the anisotropic crystal habit (lamellar crystal) from an amorphous polymer and (II) processes which increase the number of crystal nuclei. Ic and Ld are the crystal thickness and lateral dimension, respectively...

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