Entropy force

From the linear terms, the second entropy forces are... 

Kawabata44 has panted out on the basis of a simple network model that of the two derivatives, bW/blt and bW/bI2, the former should be related primarily to intramolecular forces such as the entropy force which plays a major role in the kinetic theory of rubber elasticity, while the latter should be a manifestation of intramolecular interactions. He predicted the possibility that bW/bI2 assumes negative values in the region of small defamation. In fact, the prediction was confirmed experimentally by Becker and also by the present authos. 

Such structuring is necessarily an intermolecular effect. The simplest type of an intermolecular effect, which should be treated first, is due to the crosslinks between the chains themselves. Dobson and Gordon (50) have remarked that most crosslinks are actually short chains of one or several links, which upon straining the network, become oriented but cannot be stretched. As a result an additional entropy force should arise, which has not yet been accounted for in the Gaussian theory. This force can be calculated on the basis of the Kuhn and Grun (114) chain vector orientation argument, which yields in extension... 

High-temperature species are not only identified by their vibrational spectra entropies, force fields, and molecular geometries can also be determined, particularly for small molecules (e.g., transition-metal halides MX2,...). 

A major difference of branched molecules from chain molecules is that more units are bound together and compressed into a very narrow space around the center of gravity. Hence, an immediate supposition is that in order for the monomer-monomer interaction to balance with the monomer-solvent interaction and the entropy force, and for the excluded volume effects to vanish, more attractive force between monomers are needed than is the case of chain molecules. Now we will focus our attention on concentrated systems such as non-solvent systems. An interesting idea is the influence of solvenf size on the osmotic pressure (screening effect) [19]. 

The entropy forces are weak. They are strong enough, however, to cause the retraction of the specimen when the applied load is removed. This is because (we emphasize again) the polymer chains are in the liquid state. If the tie points did not exist—if the molecules were not pinned together at particular points—the assembly would flow in a liquid-like manner. It is this combination of long-range, weak entropy forces and liquid molecules which confers on rubbers their twin properties of high deformability and, paradoxically, complete retraction (3.N.4). 

Entropy forces between physical cross-links play important roles in other drcomstances, notabty in causing ... 

The latter effect is promoted by warming, which both softens the potymer and increases the entropy forces (eqn 3.12). 

The general rule that like dissolves like bolds weU with pofymers for example, polymers absorb considerable quantities of their own monomers. If the polymer is linear it may well completely dissolve in its own monomer or other good solvent. If the solubility is lower, then the potymer, when immersed in solvent, will absorb in equilibriuin a fraction of its own mass and will not dissolve the lower the solubQity the lower the fraction. In 1.N.9 the equilibrium water contents of nylons are given for 50% and 100% RH. Hiis is the behaviour observed for vapours equilibrium absorption increases with vapour pressure. The absorbed molecules are absorbed only into the amorphous fraction of ctystalline potymers such as the nylons. Cross-linking in rubbers lowers the absorption the swelling of the network is impeded Ity the cross-links. As the network expands because of the absorption of diluent, the entropy forces between tie points increase (Figures 3.3 and 3.4) and this limits the swelling. 

The entropy force is weak. However, it is strong enough to cause a retraction in the liquid state when the applied stress is removed [18]. When the work done by the internal forces is balanced by the work done by the external forces producing the deformation, the total work, W, done for uniaxial drawing is obtained as ... 

This example illustrates how the Onsager theory may be applied at the macroscopic level in a self-consistent maimer. The ingredients are the averaged regression equations and the entropy. Together, these quantities pennit the calculation of the fluctuating force correlation matrix, Q. Diffusion is used here to illustrate the procedure in detail because diffiision is the simplest known case exlribiting continuous variables. 

Conformational Adjustments The conformations of protein and ligand in the free state may differ from those in the complex. The conformation in the complex may be different from the most stable conformation in solution, and/or a broader range of conformations may be sampled in solution than in the complex. In the former case, the required adjustment raises the energy, in the latter it lowers the entropy in either case this effect favors the dissociated state (although exceptional instances in which the flexibility increases as a result of complex formation seem possible). With current models based on two-body potentials (but not with force fields based on polarizable atoms, currently under development), separate intra-molecular energies of protein and ligand in the complex are, in fact, definable. However, it is impossible to assign separate entropies to the two parts of the complex. 

When the film thickens beyond two or three molecular layers, the effect of surface structure is largely smoothed out. It should therefore be possible, as Hill and Halsey have argued, to analyse the isotherm in the multilayer region by reference to surface forces (Chapter 1), the partial molar entropy of the adsorbed film being taken as equal to that of the liquid adsorptive. By application of the 6-12 relation of Chapter 1 (with omission of the r" term as being negligible except at short distances) Hill was able to arrive at the isotherm equation... 

Equation (3.16) shows that the force required to stretch a sample can be broken into two contributions one that measures how the enthalpy of the sample changes with elongation and one which measures the same effect on entropy. The pressure of a system also reflects two parallel contributions, except that the coefficients are associated with volume changes. It will help to pursue the analogy with a gas a bit further. The internal energy of an ideal gas is independent of volume The molecules are noninteracting so it makes no difference how far apart they are. Therefore, for an ideal gas (3U/3V)j = 0 and the thermodynamic equation of state becomes... 

The equations we have written until now in this section impose no restrictions on the species they describe or on the origin of the interaction energy. Volume and entropy effects associated with reaction (8.A) will be less if x is not too large. Aside from this consideration, any of the intermolecular forces listed above could be responsible for the specific value of x- The relationships for ASj in the last section are based on a specific model and are subject to whatever limitations that imposes. There is nothing in the formalism for AH that we have developed until now that is obviously inapplicable to certain specific systems. In the next section we shall introduce another approximation... 

As noted above, this force is counterbalanced by an entropy-based, elastic force which prevents the molecule from uncoiling. Entropy elasticity was discussed in Sec. 3.4, where the elastic (subscript el) force between crosslinks is given by Eq. (3.19) to be... 

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