Shape fluctuations

A two-dimensional (2D) molecule is a simplified abstraction because molecules have a three-dimensional (3D) form and shape. Furthermore, form and shape fluctuate, making them four-dimensional (4D) objects. Some molecular entities may be extremely flexible, others rather rigid, but a totally rigid molecule exists only at 0 K. 

Though carried further here, these theoretical ideas were first explored by Pratt and Rempe (1999). They argued that shape fluctuations were the most important concerns for simulations of biopolymers. This is particularly true for unfolded proteins. It deserves emphasis, therefore, that in this approach shape fluctuations are directly conditioned by. V 0) (JZn). 

Carbosilane dendrimers with perfluorinated end groups in perfluorohexane were studied by Stark et al. [308]. The significant deviations from simple diffusion that are observed in the NSE data in this case are attributed to shape fluctuations following a procedure that had been developed for the analysis of micro-emulsion droplet fluctuations [309]. 

Fig. 6.20 Small angle scattering intensity (triangles log I) and effective diffusion DgfKQ) obtained from g=A carbosiloxane dendrimers with perfluorinated end groups in perfluo-rohexane. The dashed line is a fit to the prediction of a model for shape fluctuations of micro-emulsion droplets, the resulting bending modulus was 0.5 k T. (Reprinted with permission from [308]. Copyright 2003 Springer Berlin Heidelberg New York)...

The question remains whether the inelastic intensity that becomes visible at the minima of the average form factor is really due to shape fluctuations or rather stems from density fluctuations (blob segmental motions) inside the dendrimer. 

Water drops become unstable and tend to break up before they reach 1 cm in diameter (see Chapter 12). Drops approaching this size show periodic shape fluctuations of relatively low amplitude (J3, M4). 

Manneville, Jan-Baptiste, Magnification of Shape Fluctuations of Active Giant Unilamellar Vesicles, 6, 351. 

The question now is whether it is observable or not. This signal has to be above the systematics (that depends on what you use to do the measurements) and above the (white) noise due to the intrinsic shape fluctuations of the source galaxies. In practice the measurement is made in the following way. Galaxy ellipticity,25000 e, are measured and a "super" pixel (say of the order of a few arcmin size) and averaged out. In the weak lensing regime we have,... 

Having established that bilayer flexibility and bilayer interaction are the mesoscopic determinants, the next question is whether these determinants can be coupled to molecular parameters. In fact, this has been done to quite some extent. In general, bilayer flexibility can be shown (both experimentally as well as theoretically by simulation methods) to be directly related to bilayer thickness, lateral interaction between heads and tails of the surfactants, type of head group (ethoxylate, sugar, etc.), type of tail (saturated, unsaturated) and specific molecular mixes (e.g. SDS with or without pen-tanol). The bilayer interaction is known to be related to characteristics such as classical electrostatics. Van der Waals, Helfrich undulation forces (stemming from shape fluctuations), steric hindrance, number, density of bilayers, ionic strength, and type of salt. Two examples will be dicussed. 

Disordered solutions of spherical micelles are not particularly viscoelastic, or even viscous, unless the volume fraction of micelles becomes high, greater than 30% by volume. Figure 12-7, for example, shows the relative viscosity (the viscosity divided by the solvent viscosity) as a function of micellar volume fraction for a solution of hydrated micelles of lithium dodecyl sulfate in water. Qualitatively, these data are reminiscent of the viscosity-volume-fraction relationship for suspensions of hard spheres, shown as a dashed line (see Section 6.2.1). The micellar viscosity is higher than that of hard-sphere suspensions because of micellar ellipsoidal shape fluctuations and electrostatic repulsions. 

Figure 12.7 Relative viscosity r)r = versus hydrated micellar volume fraction

Bending moduli can in principle be obtained for two types of systems (i) extended, flat surfaces or interfaces, the subject matter of this section, and (ii) surfaces that are already strongly curved, and for which y is zero or extremely low, such as in vesicles or micro-emulsions. For instance such moduli can be inferred from shape fluctuations, from the Kerr effect (sec. 1.7.14] or from polydispersity using some scattering technique. We repeat that this type of measurement is often ambiguous because the bending contributions to the Helmholtz energy can only be estimated when all other contributions are accurately known. 

Another force [57, 58] occurs in a multilayered system, like a swollen lamellar phase of surfactant bilayers or phospholipid vesicles. Shape fluctuations in the bilayers can give rise to steric effects that are supposed to stabilise such systems where the van der Waals and double-layer forces are very weak, as they often are. The magnitude of such fluctuations depends on the "stiffness" of die bilayer. The status of these forces is the subject of an active debate and imclear. 

These droplet shape fluctuations are governed by both k and k. The problem was theoretically treated by Milner and Safran [42-44] and values of the bending elastic constants k and i< can be obtained by using experimental methods which are able to resolve fluctuations of the surfactant layer in a microemulsion. 

In this subsection, the theoretical background for SANS and neutron spin-echo measurements carried out with o/w- and w/o-droplet microemulsions will be presented. According to Milner, Safran and others, shape fluctuations in droplet microemulsions can be described in terms of spherical harmonics [42-44]. This offers the possibility to calculate a dynamic structure factor S(q,w) or its Fourier transform, i.e. the intermediate scattering function I(q,t) for the problem, which can be used to analyse dynamical measurements by neutron spin-echo spectroscopy [45]. For the scattering from thin shells I(q,t) was calculated [43]... 

Hellweg, T., Gradzielski, M., Farago, B. and Langevin, D. (2001) Shape fluctuation of microemulsion droplets A neutron spin-echo study. Colloids Surf. A, 183-185, 159-169. 

Arleth, L. and Pedersen, J.S. (2001) Droplet polydispersity and shape fluctuations in AOT [bis(2-ethylhexyl)sulfosuccinate sodium salt] microemulsions studied by contrast variation small-angle neutron scattering. Phys. Rev. E, 63, 61406-61423. 

In Ref. [42], PEO was embedded in a w/o-droplet microemulsion and studied by small-angle neutron scattering. The authors state that this polymer does not adsorb considerably at the SDS monolayer. The important statement is that both the size polydispersity and the shape fluctuations are increased compared to the reference system without polymer. Larger shape fluctuations are also found for gelatine embedded in w/o-droplet microemulsions (see Fig. 4.10 in [43]). Here, by strong confinement, the elongated shapes... 

Another study [54] considers the molar mass dependence of droplet size fluctuations in a water-n-octane-AOT-PEO system by small-angle neutron scattering. The polydispersity increases upon increasing the molar mass and the polymer content. Usually [44], this is accompanied by larger shape fluctuations, which would make this a case 2 system. Unfortunately, shape fluctuations were not considered here. 

K. Watanabe and M. L. Klein, ]. Phys. Chem., 93, 6897 (1989). Shape Fluctuations in Ionic Micelles. 

Intramolecular interference is usually negligible for scattering from small molecules since the wavelets scattered from different segments of the same molecule all have essentially the same phase and hence add constructively at the point of observation. However, for large molecules observed at large values of q, the intramolecular interference depends on the distribution and time rate of change of molecular segmental positions. Thus these effects contain information about molecular shapes, shape fluctuations, and molecular rotations. 

Fig. 8.8.1. Bead-spring model of a flexible macromolecule. All interactions of the macromolecule with light and the surrounding medium occurs through the beads. The springs (segments) merely provide an entropic restoring force to return the beads to their equilibrium separations whenever a shape fluctuation occurs.

Further details on the dependence of shape fluctuations on temperature and... 

Figure 7 compares the shape fluctuations in deca(L-alanine) and deca(L-... 

Other methods can be used to quantify shape fluctuations and flexibility. The alternatives listed briefly here employ absolute shape descriptors ... 

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