Self-excluded volume interaction

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... 

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],... 

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. 

The above calculations assume that the gross chain conformations are those of a random walk, which is the case in the melt. However, for an isolated polymer molecule in a dilute solution, the average conformation is affected by excluded-volume interactions between one part of the chain and another. Because the chain must avoid self-intersection, the conformation of the chain will be that of a self-avoiding walk, rather than a random walk, if the solution is athermal—that is, if all interactions are negligible except excluded volume. Self-avoiding walks lead, on average, to more expanded coil dimensions, since expanded configurations are less likely than contracted ones to lead to self-intersection of the chain. Thus, in an athermal solution, the mean-square end-to-end dimension of a polymer molecule scales as... 

The upper part of the phase diagram (Fig. 5.1) corresponds to good solvents. At low concentrations, polymer coils are far from each other and behave as isolated real chains (see Section 3.3.1.1). At temperatures for which the excluded volume interaction within each chain exceeds the thermal energy kT, they begin to swell. The Flory theory prediction for the size of swollen real chains with excluded volume v > /y/N is ihs, same as the result for a self-avoiding walk of thermal blobs [Eq. (3.77)] ... 

The idea behind the replacement of the excluded volume interactions by a self-consistent potential was put forward by Edwards6 and subsequently developed by various other physicists.5 7 Concerning the critical exponent v, this method gives the same result as Flory s method (v = 3/([Pg.298]

The zero mode is the self-diffusion of the center of mass whose diffusion coefficient is given by the Stokes-Einstein relation D = k TIN. The time Tj will be proportional to the time required for a chain to diffuse an end-to-end distance, that is, R )/D = t N b lk T. This means that for time scales longer than Tj the motion of the chain will be purely diffusive. On timescales shorter than Tj, it will exhibit viscoelastic modes. However, the dynamics of a single chain in a dilute solution is more complex due to long-range forces hydrodynamic interactions between distant monomers through the solvent are present and, in good solvents, excluded volume interactions also have to be taken into account. The correction of the Rouse model for hydrodynamic interaction was done by Zimm [79]. Erom a mathematical point of view, the problem becomes harder and requires approximations to arrive at some useful results. In this case, the translational diffusion coefficient obtained is... 

Figure 2 shows an example of this 2D shape descriptor. Here, we compare two conformations of a linear polymer model. The polymer chain is an off-lattice random walk with constant bond length and excluded volume interaction between monomers (i.e., a self-avoiding walk). The constant bond length of / = 1.54 A is used to mimic poly methylene. [For a discussion and implementation of this model, see Ref. 56.]... 

The notions of functional self-similarity and scaling arise very naturally in polymer science and have found applications in this field for many years. Thus, a linear polymer chain with excluded-volume interactions is a perfect example of a physical object to which scaling should be applicable, as a subdivision of the entire chain into a collection of blobs or Kuhn macrosegments [9,14]. One of these segments may be envisioned as a fragment of sufficient length to ensure that its statistical properties are effectively independent from the remainder of the chain. 

Thus, the averaged square end-to-end separation of the iV-bond chain is related to that of a smaller X-bond fragment by the proportionality factor N/K. Bond correlations due to nonlocal (excluded-volume) interactions complicate the problem, but it is still reasonable, as we have seen, to assume that subunits of the system will, in some sense, be replicas of the entire chain. More generally, the many examples presented in the preceding sections illustrate that there are many properties of physically interesting systems that can be analyzed successfully from this point of view and that, in particular, the appropriate application of the assumption of self-similarity can be used to generate reliable approximations for physical quantities that we otherwise would be unable to calculate. 

The authors [31] introduced the notion of true self-avoiding walk (TSAW), which describes a path of random walk, restricted so as to avoid the given point visit in space with the probability, proportional to a times number, which this point was visited already. This restriction results in an excluded volume interaction reduction in comparison with SAW. The large chain compactness is a resulting effect. If in Flory approximation the exponent for SAW is given as follows [32] ... 

An important point to keep in mind is the strength of the excluded volume interaction. If one considers a strictly self-avoiding walk on a lattice (SAW) corresponding to a non-self-... 

The subscripts by Wa aJid mean the explicit dependence of the excluded volume interaction energy on the contour length cut-off, the condition r — r l > a stands for the elimination of the self-interaction inside the same segment. 

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). 

Such a repulsive contribution to the depletion interaction originates from excluded volume interactions between the depletants in case of ideal polymers and penetrable hard spheres it is absent One might expect accumulation effects also in the case of interacting polymers. From Monte Carlo simulation studies [51] and numerical self-consistent field computations [52, 53] it follows that interacting polymers do contribute to repulsive depletion interactions but with a strength of the repulsion that is nearly imperceptible. 

The self-consistent field theory has been used [39] for a terminally attached polymer on a surface with weak excluded volume interactions and high surface coverage, cr. The depletion region near the surface and the extended tail has been neglected. 

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