Finite-size deviation

As our main interest is dedicated to the expected difference of the collapse and freezing temperatures, we will focus here on the discussion of the scaling behavior of the finite-size deviation of the maximum specific-heat temperature of a finite-length polymer from the temperature, TdN) — T [127,128,130,132,133]. For polymer solutions, Floiy-Huggins mean-field theory [145] suggests... 

Every gas shows deviations from ideal behavior at high pressure. Figure 11-6 shows PV/nRT for He, F2, CH4, and N2, all of which are gases at room temperature. Notice that PV/nRT for helium increases steadily as pressure increases. Interatomic forces for helium are too small to reduce the ratio below 1, but the finite size of the helium atom generates deviations from ideality that become significant at pressures above 100 atm. 

On the other hand, extensive viscosity measurements on PB by Colby et al. [88] show that above about 300 entanglement lengths AT-dependence (the slope of /] vs. N) turns from to AP, indicating that the deviations from reptation are due to finite size effects (see also very recent results by Pyckhout et al. [89]). 

Tx, the sample size, sample number and standard deviation are related and depend on the size of the object, if this object has a finite size in correlation units. 

In order to determine the constant 7, we computed the radial distribution functions for the two types of the Sierpinski gaskets under consideration. In Fig. 6.6 these functions are plotted, as averaged over all sites of the finite gaskets at the 11th stage. Due to the finite size of the structures, deviations... 

Two specific approximations were developed to solve the problems of surface chemistry the periodic approximation, where quantum-chemical methods employ a periodic structure of the calculated system, and the cluster approximation, where the model of solid phase of finite size is created as a cutoff from the system of solid phase (it produces unsaturated dangling bonds at the border of cluster). The cluster approximation has been widely used for studying the interactions of molecules with all types of solids and their surfaces [24]. This approach is powerful in calculations of systems with deviations from the ideal periodic structure like doping and defects. Clay minerals are typical systems having such properties. 

The finite size of the nucleus leads to slighlty altered wave functions in (4) and (5) and thus to a deviation from the value given by (7). This deviation amounts... 

Figure 4 allows several conclusions First of all, the prediction of the Reissner result matches the FEM result in the small deformation region well, which shows that the finite size of the contact area does not lead to strong deviations from the analytical theory based on point like contact. This justifies an application of this result to derive elastic constants of the shell material from the measurements at small deformations. 

At pressures below 120 atmospheres nitrogen (at 0° G) shows negative deviations from ideal behavior, intermolecular attraction having a greater effect than the finite size of the molecules. 

For low surfactant concentration case, the difference in the him thickness between the hlms containing three and two micellar layers is about 15.8 nm. The experimentally observed height of the step-wise thickness transition is about 15.3 nm. For high surfactant concentration, the difference in the him thickness for the hlms with three and two micellar layers is about 8.6 nm (see Table 2) while the experimentally observed height of the step-wise thickness transition is about 10.1 nm. The increased deviation between simulation predicted hlm-thickness difference and the measured value of the step-wise thickness transition at a high surfactant concentration is probably due to the fact that our present modelling does not account for the contributions of the finite size of the small electrolyte ions as well as the solvent molecules to the electrostatic part of an effective interaction W(r) between the pair of micellar macroions. Our present model accounts for the contributions of the finite size of the electrolyte ions and the solvent molecules to the excluded volume interaction only. 

It would be rash indeed to be on the "dark flavor of the month" (except perhaps bitter chocolate). A unified dark sector proposal (77) was noted at the end of the previous section. Raman Sundrum (88) has given the graviton a finite size and urged that searches for deviations of the gravitational force from 1/R2 be continued down to the 20 pm scale. The difficulty with such experiments (89) is in getting the centers of two very... 

In the absenee of anisotropy introdueed by speeifie surfactant-surfactant interactions, a spherical droplet model is reasonable beeause it tends to minimize the surfaee energy. Deviations from spherical symmetry occru" because of the finite size and anisotropy of surfaetant moleeules and the anisotropy of interactions. Many early experimental data were interpreted on the assumption of spherieal structures. In seminal Monte Carlo studies by Haan and Pratt... 

These data are replotted in a different form in Figure 12, on the assumption that the order parameter (the coexistence density gap) for the LJ system should behave in an Ising-like manner. This is reflected in the nearly straight-line behavior of much of the data very close to the critical points the data deviate from linearity, becoming mean-field-like because of the limitation on fluctuations in a finite system. The precision of the results puts us in position to study this finite-size crossover and also other nonuniversal properties of the critical behavior of fluid phase transitions. 

We can conclude from Eq. (3.3) that the assumption that the molecules of a gas have finite size is sufficient to explain values of Z greater than unity. Apparently this size effect is the dominating one in producing deviations from ideality in hydrogen at 0 °C. It... 

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