Entropy single chain

At low concentrations, adsorption is a single-chain phenomenon. The adsorption takes place when the enthalpy gain by the monomer-surface contact with respect to the monomer-solvent contact surpasses the loss of the conformational entropy. In a good solvent the adsorption is not likely unless there is a specific interaction between monomers and the surface. At high concentrations, however, interactions between monomers dominate the free energy of the solution. The adsorption takes place when the enthalpy gain by the mono-... 

The search for the form of W of vulcanized rubbers was initiated by polymer physicists. In 1934, Guth and Mark2 and Kuhn3) considered an idealized single chain which consists of a number of links jointed linearly and freely, and derived the probability P that the end-to-end distance of the chain assumes a given value. The resulting probability function of Gaussian type was then substituted into the Boltzmann equation for entropy s, which reads,... 

The above analysis implies that the coil-globule transition is essentially a gas-liquid transition within a single chain. Unlike usual molecular gases, the translational entropy is absent due to the chain connectivity, and instead, the conformational entropy shows up. The collapsed state is a spherical droplet, that is, globule, to minimize the surface area, the size of which is self-adjusted to satisfy the mechanical balance between the inside and the outside of the globule. 

Assuming that the chains within the network obey Gaussian statistics, the entropy of a single chain whose ends are at 0,0,0 and x0>y0,z0 (see Figure 13-49) can be written (Equation 13-43) ... 

We are using a lower case s to distinguish the entropy of a single chain from that of the network as a whole, which we will designate S.) If the chain is stretched to x,y,z then its entropy becomes (Equation 13-44) ... 

This section seeks to make a quantitative evaluation of the relation between the elastic force and elongation. The calculation requires determining the total entropy of the elastomer network as a function of strain. The procedure is divided into two stages first, the calculation of the entropy of a single chain, and second, the change in entropy of a network as a function of strain. 

Since C is a constant independent of r, the variation in entropy associated with the deformation for a single chain is... 

Both theories of single-chain adsorption, described above, ignore a very important effect—the loss of conformational entropy of a trand due to its proximity to the impenetrable surface. Each adsorption blob has jb contacts with the surface and each strand of the chain near these contacts loses conformational entropy due to the proximity effect. In order to overcome this entropic penalty, the chain must gain finite energy E er per contact between a monomer and the surface. This critical energy Ecr corresponds to the adsorption transition. For ideal chains Ecr A E. The small additional free energy gain per contact kT6 should be considered in excess of the critical value Ecr,... 

The first two terms correspond to the combinatorial entropy terms of Eq. (1) and form the non-interacting part of the structure factor which is just a weighted average of the single-chain structure factors SA(q) and SB(q) of both blend components. SA(q) and SB(q) are characterized by the radius of gyration RgA= aA(NA/6)1/2 and RgB=aB(NB/6)1/2, where aA and aB are the statistical segment lengths of polymer A and polymer B, respectively. The last term of Eq. (4) yields the SANS determined interaction parameter %SANS ... 

The change in entropy of the ideal network caused by the deformation is the sum of the changes for all chains. It is shown in section 3.3.4 that the entropy of a single chain is given by... 

Using Eqs. (2.10)-(2.12) and (2.14), the entropy change associated with the single chain strand due to the deformation can be written as... 

This equation is substituted into equation 2.21 to derive the entropy of a single chain (equation 2.22) and the force required to restore a chain to its equilibrium dimensions (equation 2.21). 

Description of the mechanics of elastin requires the understanding of two interlinked but distinct physical processes the development of entropic elastic force and the occurrence of hydrophobic association. Elementary statistical-mechanical analysis of AFM single-chain force-extension data of elastin model molecules identifies damping of internal chain dynamics on extension as a fundamental source of entropic elastic force and eliminates the requirement of random chain networks. For elastin and its models, this simple analysis is substantiated experimentally by the observation of mechanical resonances in the dielectric relaxation and acoustic absorption spectra, and theoretically by the dependence of entropy on frequency of torsion-angle oscillations, and by classical molecular-mechanics and dynamics calculations of relaxed and extended states of the P-spiral description of the elastin repeat, (GVGVP) . The role of hydrophobic hydration in the mechanics of elastin becomes apparent under conditions of isometric contraction. 

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