Self-excluded volume

Two important consequences follow from the fact of the atomic. sccdc of length a being indi.stinguishable in experimentally observed quantities (Oono and Freed, 1981a Oono et al., 1981). First, the macroscopic properties do not depend on the existence of the natural minimum on the molecular scale a, so, they must be well-defined in the limit o —> 0. In the excluded volume problem of a molecular coil, the limit a —> 0 should be regarded as a rejection of the consideration how a segment interacts with itself (the self-excluded volume). 

In polymer theories, the length of a segment (molecular unit) a is the natural unit of the length scale at the molecular level, but experimentally measured quantities Q do not feel this molecular discrepancy. Hence mathematical expressions of the macroscopic quantity Q have to be well-defined in the limit a —> 0. Regarding the problem of the molecular coil excluded volume, this limit should be treated as a rejection with the consideration of the interactions of a segment with itself (self-excluded volume). 

It is well-known that the Flory mean-field theory of the polymer self-excluded volume problem yields excellent results. If R is some scalar measure of the polymer chain size, such as the radius of gyration, and N is the number of monomer units in the chain, then R scales with N as... 

The classical model for a polymer chain with self-excluded volume is the SAW. Let be the appropriate self-consistent field (SCF) for a SAW of N steps but of any size R, By analogy with the Flory argument, we are tempted to assume that for a SAW is proportional to the mean density of occupied sites,, where the volume in p (i ) has been replaced by its mean value ATva because we are considering the total ensemble of iV-step SAWs and not just the subset of size R, The SCF for a SAW must satisfy certain properties. As will be shown below, this choice for is not self-consistent for < 4. 

The intermolecular interactions screen the self-excluded volume interaction and, as a result, each correlated sequence of n monomers acts independently of the others. Thus, the mean square end-to-end of the chain will be given by... 

Polymer chains at low concentrations in good solvents adopt more expanded confonnations tlian ideal Gaussian chains because of tire excluded-volume effects. A suitable description of expanded chains in a good solvent is provided by tire self-avoiding random walk model. Flory 1151 showed, using a mean field approximation, that tire root mean square of tire end-to-end distance of an expanded chain scales as... 

Let us consider a simple self-avoiding walk (SAW) on a lattice. The net interaction of solvent-solvent, chain-solvent and chain-chain is summarized in the excluded volume between the monomers. The empty lattice sites then represent the solvent. In order to fulfill the excluded volume requirement each lattice site can be occupied only once. Since this is the only requirement, each available conformation of an A-step walk has the same probability. If we fix the first step, then each new step is taken with probability q— 1), where q is the coordination number of the lattice ( = 4 for a square lattice, = 6 for a simple cubic lattice, etc.). 

A second approach [7] allows for the effects of excluded volume correlations and self-avoidance by use of scaling arguments. In this picture, the layer is viewed... 

However, this is true only for good solvent conditions, where (c) is also the correlation length, beyond which both the excluded volume and the hydrodynamic interaction are screened and self-entanglements (intramolecular... 

Under -conditions the situation is more complex. On one side the excluded volume interactions are canceled and E,(c) is only related to the screening length of the hydrodynamic interactions. In addition, there is a finite probability for the occurrence of self-entanglements which are separated by the average distance E,i(c) = ( (c)/)1/2. As a consequence the single chain dynamics as typical for dilute -conditions will be restricted to length scales r < (c) [155,156],... 

The percolation simulations clearly allow ring formation it also represents self-avoiding statistics, i.e., includes the excluded volume effects in good solvents. Finally, the probability of placing units on lattice sites becomes more and more dependent on whether a site in the neighborhood is already occupied. In other words the percolation experiment becomes a non-mean field approach when the occupation reaches the critical percolation threshold. Therefore, strong deviations were expected between the more accurate percolation and the Flory-Stockmayer mean field approaches. Physicists were of the opinion that the mean field results must be basically wrong. 

A self-avoiding walk on a lattice is a random walk subject to the condition that no lattice site may be visited more than once in the walk. Self-avoiding walks were first introduced as models of polymer chains which took into account in a realistic manner the excluded volume effect1 (i.e., the fact that no element of space can be occupied more than once by the polymer chain). Although the mathematical problem of... 

Figures 1 a, 2a to compare with djd l and mj(n — I)a , in Figures 1 b, 2b. The limit and slopes in Figure 1 b are exact but the general pattern of behavior of the other plots is sufficiently similar to give us confidence in the conclusions. (The convergence in three dimensions is more rapid since excluded volume plays a smaller part. Similarly the self-avoiding walk approximation provides a closer fit to the correct behavior of the Ising model.)...

In his paper Domb presents a detailed analysis of the statistical properties of self-avoiding walks on lattices.1 These walks serve as models for linear polymer chains with hard-core intramolecular interactions associated with the exclusion of multiple occupancies of the lattice sites by the chain so-called chains with excluded volume. 

As a result of the branched chain architecture, TASP molecules exhibit some unique conformational properties)5 12-14 47 75 76 148 For example, the folding to a compact state proceeds via two distinct steps the onset of secondary structure in the attached peptide blocks followed by their template-directed self-assembly to a three-dimensional packing topology. Due to its characteristic branched chain connectivity, the conformational space accessible in the unfolded state is considerably reduced compared to a linear chain of similar size (excluded volume effect), resulting in a smaller chain entropy. Thus, folded TASP molecules are expected to show higher thermodynamic stability compared to unbranched polypeptides of comparable size. 

A self-consistent field theory (SCFT) for micelle formation of block copolymers in selective solvents was developed by Yuan el at. (1992). They emphasized the importance of treating the isolated chain at the same level of theoretical approximation at the micelle, in contrast to earlier approaches. This was achieved by modifying the Edwards diffusion equation for the excluded volume of polymers in solution to the case of block copolymers, with one block in a poor solvent. The results of the continuum model were compared to experimental results for PS-PI diblocks in hexadecane, which is a selective solvent for PI and satisfactory agreement was obtained. 

This is illustrated in Fig. 1,3. The. remarkable fact is that the exponent ier universal7, it is a, number independent of temperature or chemistry. Indeed, the same law is also found for self-repelling chains generated in a computer experiment. All effects of chemical micrestructure or temperature are contained in the noimniversal prefactor / . More precisely this law holds in the excluded volume limit which is reached for long chains as long as the effective interaction is repulsive. As we approach the 0-temperature, the repulsion decreases, and the chains have to be longer to reach the excluded volume law (1.3). This will be discussed in more detail below. 

We here define our model and present a self-contained introduction to perturbation theory, deriving the Feynman graph representation of the cluster expansion. To deal with solutions of finite concentration we introduce the grand-canonical ensemble and resum the cluster expansion to construct the loop expansion. We Lhen show that without further insight the expansions can be applied only in the (9-region or for concentrated solutions since they diverge term by term in the excluded volume limit. 

Leonov AI (1994) On a self-consistent molecular modelling of linear relaxation phenomena in polymer melts and concentrated solutions. J Rheol 38( 1) 1—11 Liu B, Diinweg B (2003) Translational diffusion of polymer chains with excluded volume and hydrodynamic interactions by Brownian dynamics simulation. J Chem Phys 118(17) 8061-8072... 

In this paper, a molecular thermodynamic approach is developed to predict the structural and compositional characteristics of microemulsions. The theory can be applied not only to oil-in-water and water-in-cil droplet-type microemulsions but also to bicontinuous microemulsions. This treatment constitutes an extension of our earlier approaches to micelles, mixed micelles, and solubilization but also takes into account the self-association of alcohol in the oil phase and the excluded-volume interactions among the droplets. Illustrative results are presented for an anionic surfactant (SDS) pentanol cyclohexane water NaCl system. Microstructur al features including the droplet radius, the thickness of the surfactant layer at the interface, the number of molecules of various species in a droplet, the size and composition dispersions of the droplets, and the distribution of the surfactant, oil, alcohol, and water molecules in the various microdomains are calculated. Further, the model allows the identification of the transition from a two-phase droplet-type microemulsion system to a three-phase microemulsion system involving a bicontinuous microemulsion. The persistence length of the bicontinuous microemulsion is also predicted by the model. Finally, the model permits the calculation of the interfacial tension between a microemulsion and the coexisting phase. 

Eq. (10) represents the self-consistent field equation for the local segment density of the polymer chains subject to an external electrical potential ip, a van der Waals interaction with the plates —UkT and an excluded volume interaction. Eq. (11) is a modified Poisson-Boltzmann equation in which the first term accounts for the charges of the small ions of the salt, the second term for the charges of the polyelectrolyte chains and the third one for the charges of the ions dissociated from the polyelectrolyte molecules. 

Phase diagram asymmetry can be evaluated by (i) the ratio of the biopolymer concentrations at a critical point, (ii) the angle made by the tie-lines with the concentration axis of one of the biopolymers and (iii) the length of the segment of a binodal curve between the critical point and the phase separation threshold. Association of macromolecules usually changes both their excluded volume and the affinity for the solvent water. This results in nonparallel tie-lines on the phase diagram. Normally, the tie-lines can be nonparallel since an increase in concentration of biopolymers is usually accompanied by their self-association. Equilibrium between the phases is not achievable when phase separation is accompanied by gelation. 

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