Interactions entropic repulsion

To extract a value of the step-mobility h from the grating relaxation experiments [12], we must evaluate the strength of the step-step interaction y. Computational work suggests that ydue to elastic interactions between Si(OOl) steps is 0.2 eV run [29], while, we estimate that the entropic interaction is 10 times larger. (We use a step stiffness P calculated from the geometric mean of P for Sa and Sb steps given in Ref [30] P, 0.03 eV mn-. ) Therefore, entropic repulsion should dominate, and... 

The experimental evidence for this scenario, mentioned at the outset of this section, is less compelling since it is harder to control this sort of dosage-sensitive STM experiment than a Monte Carlo simulation. (Specifically, it problematic to convert from CO dosage to the evolution time from initial instability the analysis would be better if the surface could be instantaneously de-oxidized.) From an earlier examination of the terrace-width distribution for Ag(110)2Y—>[001], Ozcomert et al. (1993) concluded that to a good approximation the step-step interactions were purely entropic repulsions (by finding a good fit to a free-fermion form (Jobs et al, 1991)). (But see also Pai et al (1994) for remarkable behavior under different conditions.) From the relationship (Bartelt el a/., 1992)... 

Direct step-step interaction terms in the step energy ( direct interactions are entropic repulsion, strain terms, electronic structure effects etc.) do influence the step fluctuations, and they also drive the spreading of step trains, wires and bumps. Nevertheless, it is instructive to first ignore these direcf step-step repulsion, as is done in... 

There is no simple, comprehensive theory and steric forces are complex and difficult to describe. Different components contribute to the force, and depending upon the situation, dominate the total force. The most important interaction is repulsive and of entropic origin. It is caused by the reduced configuration entropy of the polymer chains. If the thermal movement of a polymer chain at a surface is limited by the approach of another surface, then the entropy of the individual polymer chain decreases. In addition, the concentration of monomers in the gap increases. This leads to an increased osmotic pressure. 

A. The Thermal Fluctuations of the Interfaces for Arbitrary Interactions. After the Helfrich initial theory,18 Helfrich and Servuss17 suggested an alternate derivation of the entropic repulsion due to the confinement of a membrane between rigid walls, by considering the lipid bilayer composed of many independent pieces , whose area is related to the root mean square fluctuations of the positions of the undulatingbilayer. As shown below, this representation can be extended to interfaces interacting via arbitrary potentials. 

This expression has the same form as the heuristic approximation discussed before and the interaction decays as 1 /L. When < m < Q or 2 < m < — i.e., when the dielectric constant of the film has a value in between those of the half-spaces, the interaction is repulsive and the film will tend to thicken. In all other cases, the interaction is attractive and the film will tend to thin. We also note that the interaction energy is proportional to T, which indicates that this is a purely entropic contribution coming from the classical part of the fluctuations of the electromagnetic field. 

The second term, the free energy of inter-droplet interaction (F2), is approximated as the sum of a hard-sphere repulsive interaction (entropic) term and an attractive perturbation term. 

Here we consider the weakly adsorbed case for substrate potentials which decay (for large separations from the surface) faster than the entropic repulsion, Eq. (4), i.e., T > 1/v. This applies, e.g., to van der Waals attractive interaction between the substrate and monomers, screened electrostatic interactions, or any other short-ranged potential. In this case, fluctuations play a decisive role. In fact, for ideal chains, it can be rigorously proven (using transfer-matrix techniques) that all potentials decaying faster than z for large z have a continuous adsorption transition at a finite critical temperature T [24]. This means that flie fliickness of the adsorbed polymer layer diverges for T —> T as... 

Hard-sphere models lack a characteristic energy scale and, hence, only entropic packing effects can be investigated. A more realistic modelling has to take hard-core-like repulsion at small distances and an attractive interaction at intennediate distances into account. In non-polar liquids the attraction is of the van der Waals type and decays with the sixth power of the interparticle distance r. It can be modelled in the fonn of a Leimard-Jones potential Fj j(r) between segments... 

What are the essential features of surfactant systems An important ingredient is obviously the repulsion between water and nonpolar molecules or molecule parts, the hydrophobic force. This interaction is however highly nontrivial, and its analysis is still an active field of research [4,22,23]. Qualitatively, it is usually attributed to the strong orientational and positional correlations between nonpolar molecules in solution and the surrounding water molecules. The origin of the interaction is therefore entropic free water forms a network of hydrogen bonds. In the neighborhood... 

In Eq. (6) Ecav represents the energy necessary to create a cavity in the solvent continuum. Eel and Eydw depict the electrostatic and van-der-Waals interactions between solute and the solvent after the solute is brought into the cavity, respectively. The van-der-Waals interactions divide themselves into dispersion and repulsion interactions (Ed sp, Erep). Specific interactions between solute and solvent such as H-bridges and association can only be considered by additional assumptions because the solvent is characterized as a structureless and polarizable medium by macroscopic constants such as dielectric constant, surface tension and volume extension coefficient. The use of macroscopic physical constants in microscopic processes in progress is an approximation. Additional approximations are inherent to the continuum models since the choice of shape and size of the cavity is arbitrary. Entropic effects are considered neither in the continuum models nor in the supermolecule approximation. Despite these numerous approximations, continuum models were developed which produce suitabel estimations of solvation energies and effects (see Refs. 10-30 in 68)). 

Althongh van der Waals forces are present in every system, they dominate the disjoining pressnre in only a few simple cases, such as interactions of nonpolar and inert atoms and molecnles. It is common for surfaces to be charged, particularly when exposed to water or a liquid with a high dielectric constant, due to the dissociation of surface ionic groups or adsorption of ions from solution, hi these cases, repulsive double-layer forces originating from electrostatic and entropic interactions may dominate the disjoining pressure. These forces decay exponentially [5,6] ... 

Two repulsive contributions, osmotic and elastic contributions [31, 32], oppose the van der Waals attractive contribution where the osmotic potential depends on the free energy of the solvent-ligand interactions (due to the solvation of the ligand tails by the solvent) and the elastic potential results from the entropic loss due to the compression of ligand tails between two metal cores. These repulsive contributions depend largely on the ligand length, solvent parameters, nanopartide radius, and center-to-center distance ... 

In Equation (10), Vc represents a hard-core repulsion that is entropic in nature since it is linearly dependent on temperature in the expression for energy. Repulsion is generally associated with enthalpic interactions and we can consider the effect of an enthalpic interaction. Since Vc is associated with a single Kuhn unit we consider the average enthalpy of interaction per pair-wise interaction and the number of pair-wise interactions per Kuhn unit,... 

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