Transition fluctuation forces

The interest in vesicles as models for cell biomembranes has led to much work on the interactions within and between lipid layers. The primary contributions to vesicle stability and curvature include those familiar to us already, the electrostatic interactions between charged head groups (Chapter V) and the van der Waals interaction between layers (Chapter VI). An additional force due to thermal fluctuations in membranes produces a steric repulsion between membranes known as the Helfrich or undulation interaction. This force has been quantified by Sackmann and co-workers using reflection interference contrast microscopy to monitor vesicles weakly adhering to a solid substrate [78]. Membrane fluctuation forces may influence the interactions between proteins embedded in them [79]. Finally, in balance with these forces, bending elasticity helps determine shape transitions [80], interactions between inclusions [81], aggregation of membrane junctions [82], and unbinding of pinched membranes [83]. Specific interactions between membrane embedded receptors add an additional complication to biomembrane behavior. These have been stud-... 

So far, the solvent coordinate has not been defined. As noted at the beginning of this Section, the time dependent friction is to be found for the reacting solute fixed at the transition state value x of x. By (3.14), its dynamics were related to those of an (unspecified) solvent coordinate. v. One strategy to identify the solvent coordinate, its frequency, friction, etc., would be to derive an equation of motion for the relevant fluctuating force SF there. To this end, one can use a double-membered projection technique in terms of 8F and 8F. In particular, we define the projection operator... 

Let us make some connections to the results which came from the previous model development. First, if we compare (3.19)-(3.22) with (3.11)-(3.15), a natural identification of the solvent coordinate s in Sec. 3 is in fact just the fluctuating force SF on x at the transition state. (Note especially that this choice associates the solvent coordinate with a direct measure of the relevant solute-solvent interaction.) The solvent mass, force constant and frequencies in Sec. 3 would then be given molecular expressions via (3.19)-(3.21), while the solvent friction i (t) of Sec. 3 would be the friction per mass for Sf (3.22),... 

It is of great interest to obtain molecular-level descriptions of the structural fluctuations occurring in solutions phase systems. In this section we discuss how a solute vibrational transition is influenced by the fluctuating forces... 

Some representative examples of common zero-temperature VER mechanisms are shown in Fig. 2b-f. Figures 2b,c describe the decay of the lone vibration of a diatomic molecule or the lowest energy vibrations in a polyatomic molecule, termed the doorway vibration (63), since it is the doorway from the intramolecular vibrational ladder to the phonon bath. In Fig. 2b, the excited doorway vibration 2 lies below large molecules or macromolecules. In the language of Equation (4), fluctuating forces of fundamental excitations of the bath at frequency 2 are exerted on the molecule, inducing a spontaneous transition to the vibrational ground state plus excitation of a phonon at Fourier transform of the force-force correlation function at frequency 2, denoted C( 2). 

The stochastic model of ion transport in liquids emphasizes the role of fast-fluctuating forces arising from short (compared to the ion transition time), random interactions with many neighboring particles. Langevin s analysis of this model was reviewed by Buck [126] with a focus on aspects important for macroscopic transport theories, namely those based on the Nernst-Planck equation. However, from a microscopic point of view, application of the Fokker-Planck equation is more fruitful [127]. In particular, only the latter equation can account for local friction anisotropy in the interfacial region, and thereby provide a better understanding of the difference between the solution and interfacial ion transport. 

An ordering phase transition is characterized by a loss of symmetry the ordered phase has less symmetry than the disordered one. Hence, an ordering process leads to the coexistence of different domains of the same ordered phase. An interface forms whenever two such domains contact. The thermodynamic behavior of this interface is governed by different forces. The presence of the underlying lattice and the stability of the ordered domains tend to localize the interface and to reduce its width. On the other hand, thermal fluctuations favor an interfacial wandering and an increase of the interface width. The result of this competition depends strongly on the order of the bulk phase transition. 

The FvdM as well as the BMVW model neglects thermal fluctuation effects both are T = 0 K theories. Pokrovsky and Talapov (PT) have studied the C-SI transition including thermal effects. They found that, for T 0 K the domain walls can meander and collide, giving rise to an entropy-mediated repulsive force of the form F where I is the distance between nearest neighbor walls. Because of this inverse square behavior, the inverse wall separation, i.e. the misfit m, in the weakly incommensurate phase should follow a power law of the form... 

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