Real chain free energy

A simpler, analytic approach is to employ the standard idea of replacing the real system with a computationally tractable reference system (polymer chain model in our case), the parameters of which are variation-ally optimized using an approximate single-chain free energy expression. Two such schemes have been explored that differ in both the choice of reference chain model and the form of the approximate free energy. 

AG is always negative, and the decrease in free energy can be due to adsorption effects (change in AH) or entropic interactions (change in AS). AS is always operating when the polymer chain cannot occupy all possible conformations in a pore (confined space) due to the limited size of the pore relative to the size of the macromolecule. In a real... 

As it follows from [14] in the field of the chains intertwining the molar free energy of the conformation is linear function of relative concentration of macromolecules and is described by the following expression in approximation by deformation of m-ball in real solution... 

VII. Multi-parameter Correlation Equations All authors who have seriously considered the scope and limitations of the linear free-energy relationships have recognized the existence of real deviations. Frequently, the limitations of the Hammett eq. (1) for certain substituents in certain situations were considered to be indicative of a duality of u-constants. Hammett noticed that the reactions of anilines and phenols required a special value for aJt NOt, 1.27, in contrast to the value, 0.778, derived from benzoic acids. An example is the increased acidity of p-nitrophenol over that expected on the basis of the a constant based on benzoic acid. Resonance interaction between the substituent and the side-chain is presumed to be responsible ... 

Classical density functional theory (DFT) [18,19] treats the cluster formation free energy as a functional of the average density distributions of atoms or molecules. The required input information is an intermolecular potential describing the substances at hand. The boundary between the cluster and the surrounding vapor is not anymore considered sharp, and surface active systems can be studied adequately. DFT discussed here is not to be confused with the quantum mechanical density functional theory (discussed below), where the equivalent of the Schrodinger equation is expressed in terms of the electron density. Classical DFT has been used successfully to uncover why and how CNT fails for surface active systems using simple model molecules [20], but it is not practically applicable to real atmospheric clusters if the molecules are not chain-like, the numerical solution of the problem gets too burdensome, unless the whole molecule is treated in terms of an effective potential. 

The Flory estimate of the entropic contribution to the free energy of a real chain is the energy required to stretch an ideal chain to end-to-end distance 7 [Eq. (2.101)] ------------------------------------------------------... 

The total free energy of a real chain in the Flory approximation is the sum of the energetic interaction and the entropic contributions ... 

The minimum free energy of the chain (obtained by setting dFjdR = Q) gives the optimum size of the real chain in the Flory... 

Ro and are the end-to-end distances of unconfined ideal and real chains, respectively. These calculations can be generalized to confinement a polymer with fractal dimension lju from its original size bJST to a cylinder with diameter D. The confinement free energy in this case is (derived in Problem 3.16)... 

Minimizing this free energy with respect to R gives the size of a real chain between plates of spacing D ... 

Notice that if Eqs (3.69) and (3.70) are blindly substituted into Eqs (3.67) and (3.68), the conclusion would be that the adsorption free energy is zero for both ideal and real chains. This exemplifies the disadvantage of scaling calculations. There are unspecified prefactors of order unity in both terms of Eqs (3.67) and (3.68), which invalidates the blind substitution. 

The free energy of stretching a real linear chain in a good solvent has a stronger dependence on size R than the quadratic dependence of the ideal chain ... 

Calculate the free energy for compressing a real chain into a cylindrical tube with diameter D. Assume an arbitrary scaling exponent u in the dependence of end-to-end distance of the chain on the number of monomers R = bN. ... 

Consider a real chain adsorbed at a surface with an excess free energy gain per monomer SkT. Assume that the monomer concentration decreases as a... 

At what thickness of the slit D does the free energy change form between real and ideal chain expressions ... 

The free energy of a Ginzburg-Landau field describing a system of weakly coupled chains in a plane is identified with the ground-state energy of a linear array of quantum anharmonic oscillators. The equivalent Hamiltonian is simplified for both the real and complex fields using a truncated basis of states. Results for both the real and complex fields will be discussed. In addition, the behavior of the specific heat and inverse correlation length for finite numbers of weakly coupled chains will be discussed. 

The real liquid crystalline polymers exhibit finite flexibility. This kind of polymer was studied extensively by Khokhlov and his co-workers. Assume that the semiflexible chain has the total contour length L, and Kuhn length l and diameter D, and L l D. Analogous to the Onsager model, the free energy of the polymer in solution is composed of the conformational... 

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