Excluded-volume interactions

Another interesting version of the MM model considers a variable excluded-volume interaction between same species particles [92]. In the absence of interactions the system is mapped on the standard MM model which has a first-order IPT between A- and B-saturated phases. On increasing the strength of the interaction the first-order transition line, observed for weak interactions, terminates at a tricritical point where two second-order transitions meet. These transitions, which separate the A-saturated, reactive, and B-saturated phases, belong to the same universality class as directed percolation, as follows from the value of critical exponents calculated by means of time-dependent Monte Carlo simulations and series expansions [92]. 

FIQ. 1 Sketch of the BFM of polymer chains on the three-dimensional simple cubic lattice. Each repeat unit or effective monomer occupies eight lattice points. Elementary motions consist of random moves of the repeat unit by one lattice spacing in one lattice direction. These moves are accepted only if they satisfy the constraints that no lattice site is occupied more than once (excluded volume interaction) and that the bonds belong to a prescribed set of bonds. This set is chosen such that the model cannot lead to any moves where bonds should intersect, and thus it automatically satisfies entanglement constraints [51],... 

Fig. 21(a) that, in contrast to the case of polymer melts where the screening of excluded volume interactions leads to 1/2, in the porous medium... 

Other questions which will have to be studied in more detail refer to the role of micelle interactions. Note that they are not included in the approximate analysis sketched above. Von Gottberg et al. [142] have argued that excluded volume interactions between micelles are responsible for the shght... 

One of the signatures of densely tethered chains is expressed in Eq. 5, namely the linear variation of L with N. This stands in marked contrast with free chains where excluded volume interaction produces, at most, an R N3/5 distortion from the R N1/2 unperturbed dimensions. Tethered layers are stretched and this is the origin of their interesting behavior. 

The simplest situation is the symmetrical one (NA = NB), with the solvent equally good for both blocks. We imagine that the excluded volume interactions of A and B are stronger than the A-B repulsive interactions so that the overall structure of the layer is like that of a single component in other words, both components are equally stretched. The issue is whether or not they are homogeneously mixed with one another in the monolayer. This is essentially a two-dimensional random mixing process. In that spirit, we write the free energy... 

Some years ago, on the basis of the excluded-volume interaction of chains, Hess [49] presented a generalized Rouse model in order to treat consistently the dynamics of entangled polymeric liquids. The theory treats a generalized Langevin equation where the entanglement friction function appears as a kernel... 

On macroscopic length scales, as probed for example by dynamic mechanical relaxation experiments, the crossover from 0- to good solvent conditions in dilute solutions is accompanied by a gradual variation from Zimm to Rouse behavior [1,126]. As has been pointed out earlier, this effect is completely due to the coil expansion, resulting from the presence of excluded volume interactions. 

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],... 

It is generally accepted that in semi-dilute solutions under good solvent conditions both the excluded volume interactions and the hydrodynamic interactions are screened owing to the presence of other chains [4,5,103], With respect to the correlation lengths (c) and H(c) there is no consensus as to whether these quantities have to be equal [11] or in general would be different [160],... 

Note that packing contributions are most obvious in the outer shell contribution to Eq. (33) because the indicator function 0 (1 — bj) constrains that term to the case where no occupancy of the defined inner shell is permitted. These excluded volume interactions are the essense of packing contributions. To study those contributions, we consider a fictitious solute that does not interact with the solvent at all e AU/kT = 1. In that case, of course, p7x is zero and we write... 

Vlugt, T. J. H., Measurement of chemical potentials of systems with strong excluded volume interactions by computing the density of states, Mol. Phys. 2002, 100, 2763-2771... 

In an ensemble of flexible polymer chains, the instantaneous separation of two segments i and j varies from one molecule to another. Ensemble averages such as required in Eq. 2 are obtained by specifying W(r-jj), the probability that segments i and j are separated by ry. In an elastomeric rubber which is not so highly swollen that excluded volume interactions become important, and which is not too greatly deformed, W(r- j) takes a particularly simple form... 

A problem arises, in that the strong r b dependence of My requires that close overlap of spins be prevented. Thus, even though excluded volume interactions have no effect on chain dimensions in the bulk amorphous phase, it is important in the present application to build in an excluded volume effect (simulated with appropriate hard sphere potentials), so that occasional close encounters of the RIS phantom segments do not lead to unrealistically large values of M2. 

Our conclusions are that a) theta conditions are not necessary to obtain experimental mK values unperturbed by excluded volume interactions, b) the RIS and BAA approximations are applicable and c) mK is a very sensitive to the details of the RIS model, the tacticity and the composition of a copolymer. 

In scenario (c) corresponding to semidilute solutions, polyelectrolyte chains interpenetrate. Under these circumstances, there are three kinds of screening. The electrostatic interaction, excluded volume interaction, and the hydro-dynamic interaction between any two segments of a labeled polyelectrolyte chain are all screened by interpenetrating chains. Each of these three interactions is associated with a screening length, namely, and These screening... 

N is the total number of monomers, (p the polymer volume fraction and Pi and Pi/2 the form factors of the total copolymer and of the single blocks respectively. 12=Vd=Vh is the excluded volume interaction parameter which relates to the second virial coefficient A2=vN/ 2Mc). 

---