Short-range predictions

Even if short-range prediction schemes led to 100% predictability of the conformational states of the residues of a protein, the structure would not resemble the correct one (Burgess and Scheraga, 1975a). Long-range interactions are required in order that the molecule achieve its correct shape. Also, if energy minimization is started before the molecular shape is approximately correct, then minimization will be trapped in a non-native local... 

The integral under the heat capacity curve is an energy (or enthalpy as the case may be) and is more or less independent of the details of the model. The quasi-chemical treatment improved the heat capacity curve, making it sharper and narrower than the mean-field result, but it still remained finite at the critical point. Further improvements were made by Bethe with a second approximation, and by Kirkwood (1938). Figure A2.5.21 compares the various theoretical calculations [6]. These modifications lead to somewhat lower values of the critical temperature, which could be related to a flattening of the coexistence curve. Moreover, and perhaps more important, they show that a short-range order persists to higher temperatures, as it must because of the preference for unlike pairs the excess heat capacity shows a discontinuity, but it does not drop to zero as mean-field theories predict. Unfortunately these improvements are still analytic and in the vicinity of the critical point still yield a parabolic coexistence curve and a finite heat capacity just as the mean-field treatments do. 

Altematively, tire polymer layers may overlap, which increases tire local polymer segment density, also resulting in a repulsive interaction. Particularly on close approach, r < d + L, a steep repulsion is predicted to occur. Wlren a relatively low molecular weight polymer is used, tire repulsive interactions are ratlier short-ranged (compared to tire particle size) and the particles display near hard-sphere behaviour (e.g., [11]). 

In the above analysis, Johnson et al. [6] assume that the interfacial forces act only when the surfaces are in contact, i.e. the attractive forces are considered to be of infinitesimally short range. This analysis ignores the forces acting just outside the edge of the contact circle. Because of this, the theory predicts an infinite tensile stress at the edge of the contact. If the attractive force between the surfaces is allowed to have a finite range, the infinity in the tensile stress disappears. The stress at the edge of the contact circle is still tensile, but it remains finite. 

Incomplete Dissociation into Free Ions. As is well known, there are many substances which behave as a strong electrolyte when dissolved in one solvent, but as a weak electrolyte when dissolved in another solvent. In any solvent the Debye-IIiickel-Onsager theory predicts how the ions of a solute should behave in an applied electric field, if the solute is completely dissociated into free ions. When we wish to survey the electrical conductivity of those solutes which (in certain solvents) behave as weak electrolytes, we have to ask, in each case, the question posed in Sec. 20 in this solution is it true that, at any moment, every ion responds to the applied electric field in the way predicted by the Debye-Hiickel theory, or does a certain fraction of the solute fail to respond to the field in this way In cases where it is true that, at any moment, a certain fraction of the solute fails to contribute to the conductivity, we have to ask the further question is this failure due to the presence of short-range forces of attraction, or can it be due merely to the presence of strong electrostatic forces ... 

The division of the interface into an inner layer and a diffuse layer has been a matter of discussion in view of the molecular dimensions of the inner layer.122-126,279-285 However, the contribution of a constant capacitance is an experimental fact. Furthermore, molecular theories for electrolytes near a charged hard wall282 as well as phenomenological nonlocal electrostatic theories283 predict such a component without artificial introduction of any inner layers. This turns out to be an effect of the short-range structure of the solvent.279-285... 

Lattice gas models are simple to construct, but the gross approximations that they involve mean that their predictions must be treated with care. There are no long-range interactions in the model, which is unrealistic for real molecules the short-range interactions are effectively hard-sphere, and the assumption that collisions lead to a 90° deflection in the direction of movement of both particles is very drastic. At the level of the individual molecule then, such a simulation can probably tell us nothing. However, at the macroscopic level such models have value, especially if a triangular or hexagonal lattice is used so that three-body collisions are allowed. 

Another very complicated problem where the approach to equilibrium with time after a quenching experiment is described by an asymptotic law is the owth of wetting layers, in a situation where thermal equilibrium would require the surface to be coated with a macroscopically thick film, but is initially nonwet. For a short-range surface potoitial as discussed in section 3.5, analytical theories predict for a non-conserved density a growth of the thicknm of the layer according to a law f(t) oc In t, and this has in fact been observed by simulations . In the case where the surface potential decays with stance z from the surface as z, the prediction for the thickness l(t) is for the nonconserved case and... 

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