Chain chemical potential

Fig. 23. Typical interaction potential energy vs. gap half-width (scaled on Rt) for restricted equilibrium (curve C) calculated by Ploehn (1988) using the Flory-Huggins SCF of Eq. (70). The components of the potential are the potential for full equilibrium, i.e., the surface tension, (curve A) and the chains chemical potential (curve B). Parameters are x = 1, x = 0.488, and n= 1129 the dosage

When do we cross over from the weak penetration regime of eq. (III.57) to the strong penetration regime for concentrated systems A detailed answer to this question could be obtained from a study of the chain chemical potential, but the essential features may be reached more simply. Let us start from a tube with a diameter D somewhat larger than the correlation length in the bulk. Then we expect to have a depletion layer of thickness near the tube walls, as described in Section ni.3.1. The perturbing effects of the wall do not extend further than... 

Gibbs free energy density of the phase of A/B-chains Gibbs free energy of a polymer chain chemical potential of a monomer in the melt chemical potential of a monomer in an infinite crystal difference in the chemical potential between phases i and j (= 9j - 9i)... 

Pablo J J, M Laso, and U W Suter 1992. Estimation of the Chemical Potential of Chain Molecules by Simulation. Journal of Chemical Physics 96 6157-6162. jkstra M. 1997. Confined Thin Films of Linear and Branched Alkanes. Journal of Chemical Physics 107 3277-3288. 

Hydrophobic fibers are difficult to dye with ionic (hydrophilic) dyes. The dyes prefer to remain in the dyebath where they have a lower chemical potential. Therefore nonionic, hydrophobic dyes are used for these fibers. The exceptions to the rule are polyamide and modified polyacrylonitriles and modified polyester where the presence of a limited number of ionic groups in the polymer, or at the end of polymer chains, makes these fibers capable of being dyed by water-soluble dyes. 

Internal and External Phases. When dyeing hydrated fibers, for example, hydrophUic fibers in aqueous dyebaths, two distinct solvent phases exist, the external and the internal. The external solvent phase consists of the mobile molecules that are in the external dyebath so far away from the fiber that they are not influenced by it. The internal phase comprises the water that is within the fiber infrastmcture in a bound or static state and is an integral part of the internal stmcture in terms of defining the physical chemistry and thermodynamics of the system. Thus dye molecules have different chemical potentials when in the internal solvent phase than when in the external phase. Further, the effects of hydrogen ions (H" ) or hydroxyl ions (OH ) have a different impact. In the external phase acids or bases are completely dissociated and give an external or dyebath pH. In the internal phase these ions can interact with the fiber polymer chain and cause ionization of functional groups. This results in the pH of the internal phase being different from the external phase and the theoretical concept of internal pH (6). 

Eq. (5) is useful when analyzing different approximations in the theory of inhomogeneous fluids. In particular, if all the terms involving third- and higher-order correlations in the right-hand side of Eq. (5) are neglected, and if Pi(ro))P2( o)i )Pv( o) are chosen as the densities of species for a uniform system at temperature T and the chemical potentials p,, the singlet hypemetted chain equation (HNCl) [50] results... 

Let us describe the results for the case of a matrix made of chains with four beads, i.e., m = M = 4. We observe that both the PY and the HNC approximation quite accurately describe the adsorption isotherms in matrices of different microporosity (Fig. 5). However, a discrepancy between the HNC results and simulation data increases with increasing chemical potential, especially at higher matrix densities. The structural... 

We close these introductory remarks with a few comments on the methods which are actually used to study these models. They will for the most part be mentioned only very briefly. In the rest of this chapter, we shall focus mainly on computer simulations. Even those will not be explained in detail, for the simple reason that the models are too different and the simulation methods too many. Rather, we refer the reader to the available textbooks on simulation methods, e.g.. Ref. 32-35, and discuss only a few technical aspects here. In the case of atomistically realistic models, simulations are indeed the only possible way to approach these systems. Idealized microscopic models have usually been explored extensively by mean field methods. Even those can become quite involved for complex models, especially for chain models. One particularly popular and successful method to deal with chain molecules has been the self-consistent field theory. In a nutshell, it treats chains as random walks in a position-dependent chemical potential, which depends in turn on the conformational distributions of the chains in... 

From the standpoint of thermodynamics, the essential quantity which governs the course of a chemical reaction is the chemical potential of the system or, in the case of interest, the free energy storage within the molecular coil. This quantity, unfortunately, is difficult to evaluate in non-steady flow. At modest extension ratios (X < 4), the free energy storage of a freely-jointed bead-spring chain is... 

Each of the respiratory chain complexes I, III, and IV (Figures 12-7 and 12-8) acts as a proton pump. The inner membrane is impermeable to ions in general but particularly to protons, which accumulate outside the membrane, creating an electrochemical potential difference across the membrane (A iH )-This consists of a chemical potential (difference in pH) and an electrical potential. 

It will be observed that the chemical potentials depend appreciably on the chain length, represented by x only at low concentrations. The influence of the chain length vanishes with increase in concentration of... 

Electrochemical redox studies of electroactive species solubilized in the water core of reverse microemulsions of water, toluene, cosurfactant, and AOT [28,29] have illustrated a percolation phenomenon in faradaic electron transfer. This phenomenon was observed when the cosurfactant used was acrylamide or other primary amide [28,30]. The oxidation or reduction chemistry appeared to switch on when cosurfactant chemical potential was raised above a certain threshold value. This switching phenomenon was later confirmed to coincide with percolation in electrical conductivity [31], as suggested by earlier work from the group of Francoise Candau [32]. The explanations for this amide-cosurfactant-induced percolation center around increases in interfacial flexibility [32] and increased disorder in surfactant chain packing [33]. These increases in flexibility and disorder appear to lead to increased interdroplet attraction, coalescence, and cluster formation. 

Although many different processes can control the observed swelling kinetics, in most cases the rate at which the network expands in response to the penetration of the solvent is rate-controlling. This response can be dominated by either diffu-sional or relaxational processes. The random Brownian motion of solvent molecules and polymer chains down their chemical potential gradients causes diffusion of the solvent into the polymer and simultaneous migration of the polymer chains into the solvent. This is a mutual diffusion process, involving motion of both the polymer chains and solvent. Thus the observed mutual diffusion coefficient for this process is a property of both the polymer and the solvent. The relaxational processes are related to the response of the polymer to the stresses imposed upon it by the invading solvent molecules. This relaxation rate can be related to the viscoelastic properties of the dry polymer and the plasticization efficiency of the solvent [128,129],... 

Widom test-particle method. Provides the chemical potential in various ensembles. Relatively easy to implement and can be used as an additional measurement in standard MC ensembles (and also MD). Computational overhead is small. Yields good accuracy in simple systems, although less reliable in very dense or complex systems (i.e., chain molecules). 

One of the interests in confined polymers arises from adsorption behavior— that is, the intake or partitioning of polymers into porous media. Simulation of confined polymers in equilibrium with a bulk fluid requires simulations where the chemical potentials of the bulk and confined polymers are equal. This is a difficult task because simulations of polymers at constant chemical potential require the insertion of molecules into the fluid, which has poor statistics for long chains. Several methods for simulating polymers at constant chemical potential have been proposed. These include biased insertion methods [61,62], novel simulation ensembles [63,64], and simulations where the pore is physically connected to a large bulk reservoir [42]. Although these methods are promising, so far they have not been implemented in an extensive study of the partitioning of polymers into porous media. This is a fruitful avenue for future research. 

The change of chemical potential due to the elastic retractive forces of the polymer chains can be determined from the theory of rubber elasticity (Flory, 1953 Treloar, 1958). Upon equaling these two contributions an expression for determining the molecular weight between two adjacent crosslinks of a neutral hydrogel prepared in the absence of... 

Peppas and Merrill (1977) modified the original Flory-Rehner theory for hydrogels prepared in the presence of water. The presence of water effectively modifies the change of chemical potential due to the elastic forces. This term must now account for the volume fraction density of the chains during crosslinking. Equation (4) predicts the molecular weight between crosslinks in a neutral hydrogel prepared in the presence of water. 

The next problem is to find an expression for Asg. This entropy difference is a function of the particle volume fractions in the dispersion ( ) and in the floe (<(> ). As a first approximation, we assume that Ass is independent of the concentration and chain length of free polymer. This assumption is not necessarily true the floe structure, and thus < >f, may depend on the latter parameters because also the solvent chemical potential in the solution (affected by the presence of polymer) should be the same as that in the floe phase (determined by the high particle concentration). However, we assume that these effects will be small, and we take as a constant. 

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