Excluded volume expansion

Theories of the excluded volume expansion factor for polymers have a long history, and as discussed elsewhere, this problem led to some of the earliest applications of computer models to polymers. More recently, Krishnaswamy and Fixman have used simulations to verify a theoretical treatment of the expansion factor for polyelectrolytes (251). 

A number of refinements and applications are in the literature. Corrections may be made for discreteness of charge [36] or the excluded volume of the hydrated ions [19, 37]. The effects of surface roughness on the electrical double layer have been treated by several groups [38-41] by means of perturbative expansions and numerical analysis. Several geometries have been treated, including two eccentric spheres such as found in encapsulated proteins or drugs [42], and biconcave disks with elastic membranes to model red blood cells [43]. The double-layer repulsion between two spheres has been a topic of much attention due to its importance in colloidal stability. A new numeri-... 

Theta conditions in dilute polymer solutions are similar to tire state of van der Waals gases near tire Boyle temperature. At this temperature, excluded-volume effects and van der Waals attraction compensate each other, so tliat tire second virial coefficient of tire expansion of tire pressure as a function of tire concentration vanishes. On dealing witli solutions, tire quantity of interest becomes tire osmotic pressure IT ratlier tlian tire pressure. Its virial expansion may be written as... 

Our primary interest in the Flory-Krigbaum theory is in the conclusion that the second virial coefficient and the excluded volume depend on solvent-solute interactions and not exclusively on the size of the polymer molecule itself. It is entirely reasonable that this should be the case in light of the discussion in Sec. 1.11 on the expansion or contraction of the coil depending on the solvent. The present discussion incorporates these ideas into a consideration of solution nonideality. 

The parameter a which we introduced in Sec. 1.11 to measure the expansion which arises from solvent being imbibed into the coil domain can also be used to describe the second virial coefficient and excluded volume. We shall see in Sec. 9.7 that the difference 1/2 - x is proportional to. When the fully... 

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]. 

The quantity b has the dimension of a volume and is known as the excluded volume or the binary cluster integral. The mean force potential is a function of temperature (principally as a result of the soft interactions). For a given solvent or mixture of solvents, there exists a temperature (called the 0-temperature or Te) where the solvent is just poor enough so that the polymer feels an effective repulsion toward the solvent molecules and yet, good enough to balance the expansion of the coil caused by the excluded volume of the polymer chain. Under this condition of perfect balance, all the binary cluster integrals are equal to zero and the chain behaves like an ideal chain. 

The analogy with the virial expansion of PF for a real gas in powers of 1/F, where the excluded volume occupies an equivalent role, is obvious. If the gas molecules can be regarded as point particles which exert no forces on one another, u = 0, the second and higher virial coefficients (42, Azy etc.) vanish, and the gas behaves ideally. Similarly in the dilute polymer solutions when w = 0, i.e., at 1 = , Eqs. (70), (71), and (72) reduce to vanT Hoff s law... 

In the classical theories of polyelectrolytes, the chain expansion is characterized by the electrostatic excluded volume parameter, zel with ... 

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. 

A second problem with the random walk model concerns the interaction between segments far apart along the contour of the chain but which are close together in space. This is the so-called "excluded volume" effect. The inclusion of this effect gives rise to an expansion of the chain, and in three-dimensions, 2 a, r3/5 (9), rather than the r dependence given in equation (I). 

Expansion of Thickness of the Adsorbed Layer. In the low salt concentration the large thickness compared with the case of the Theta solvent (4.17 M NaCl) is considered to be due to the electrostatic repulsion, i.e., the excluded volume effect of the adsorbed NaPSS chains. Usually, the expansion factor at, defined by the ratio of the thickness in good solvent and that in the Theta solvent, is used to quantitatively evaluate the excluded volume effect for the adsorbed polymers. 

For adsorption of nonionic polymer, Hoeve (15) and Jones-Richmond (16) attempted to incorporate the excluded-volume effect into the expansion factor, respectively. They suggested that the thickness of the adsorbed layer in good and 0 solvents should be taken at the same adsorbance and molecular weight3 respectively. We may calculate the expansion factor at the bulk NaPSS concentration of 0.02 g/lOOml, since the adsorbances are almost the same for the respective NaCl concentrations, as seen from Figure 5. 

Fig. 11. A log-log plot of the expansion factor, a, vs reduced excluded volume, z% for MC data of 12-arms stars with N=25-109 units triangles N=25 crosses N=49 asterisks N=85 squares N=109. The top and bottom solid lines and figures represent slopes corresponding to the predicted asymptotic behaviors for the EV and sub-theta regimes, respectively. Reprinted with permission from [143]. Copyright (1992) American Chemical Society...

Lipson et al. [181] performed an MC study on different types of lattices for three functional comb chains with two branched points, or H-combs, in the excluded volume regime. The variation of the branch mean size with its length follows the expected scaling law in terms of critical exponent, Rg =Nj, . This is in accordance with the expected behavior in the low branching (or mushroom) regime, and it is also in agreement with RG calculations [182]. In the Lipson et al. simulations, expansion of the different branches was analyzed by evaluating their ampHtudes in this power-law. Thus, the internal branches (backbone seg-... 

The size of a polymer molecule in solution is influenced by both the excluded volume effect and thermodynamic interactions between polymer segments and the solvent, so that in general =t= . The Flory (/S) expansion factor a is introduced to express this effect, by writing ... 

The size of molecules in solution, measured by , is important both in its own right and because of the effects of changes in on interactions between molecules in solution. For linear polymers, the expansion factor a of Eq. (3.2) can be expressed as a power series in a dimensionless parameter z which is related to the excluded volume per segment pair, p, and the molecular size ... 

Here a0 is a constant called the effective bond length of the chain, and as(z) is a dimensionless quantity called the linear expansion factor of the chain. The latter depends on long-range interactions between pairs of monomer units and chain length through the so-called excluded-volume parameter z. For details of these quantities characterizing the dimensions of random-coil polymers, the reader is referred to a recently published book by Yamakawa (40). At this place we simply note that as tends to unity in the absence of excluded-volume effect. 

Expansion of subchains in a linear PE chain with excluded volume is evaluated by two methods. (1) Monte-Carlo chains with methylene groups participating in long-range interactions behave as impenetrable spheres with a diameter of 300 pm, and (25 generator matrix calculations in which expansion is produced without any effect on the probability of a trans placement in an infinitely long chain. 

The influence of the chain expansion produced by excluded volume on the mean-square optical anisotropy is studied in six types of polymers (PE, PVC, PVB, PS, polylp-chlorostyrene), polylp-bromostyrenel. RIS models are used for the configuration statistics of the unperturbed chains. The mean-square optical anisotropy of PE is found to be insensitive to excluded volume. The mean-square optical anisotropy of the five other polymers, on the other hand, is sensitive to the imposition of the excluded volume if the stereochemical composition is exclusively racemic. Much smaller effects are seen in meso chains and in chains with Bernoullian statistics and an equal probability for meso and racemic diads. 

Generator matrix methods are used to compute g, 0, and cp2 > 0 for PVB, PVC, and PS chains as function of the stereochemical composition. Simulations that permit introduction of excluded volume show that for all three chains Is insensitive to /0 unless the stereochemical composition is predominantly racemic. The response of to chain expansion is more dramatic in racemic PVC than in the other two polymers. 

Expansion is considered for finite, regular polyethylene stars perturbed by the excluded volume effect. An RIS model is used for the chain statistics. The number of bonds in each branch ranges up to 10 240, and the functionality of the branch point ranges up to 20. The form of the calculation employed here provides a lower bound for the expansion. If the number, n, of bonds in the polymers is heid constant, expansion is found to decrease with increasing branch point functionality. Two factors dictate the manner in which finite stars approach the limiting behavior expected for very large stars, These two factors are the chain length dependence at small n of the characteristic ratio and of fa -a3) / n1/2. 

For dilute solutions in good solvents the net excluded volume is positive, and coil dimensions are expanded beyond their unperturbed values. The expansion... 

In Fiery s theory of the excluded volume (27), the chains in undiluted polymer systems assume their unperturbed dimensions. The expansion factor in solutions is governed by the parameter (J — x)/v, v being the molar volume of solvent and x the segment-solvent interaction (regular solution) parameter. In undiluted polymers, the solvent for any molecule is simply other polymer molecules. If it is assumed that the excluded volume term in the thermodynamic theory of concentrated systems can be applied directly to the determination of coil dimensions, then x is automatically zero but v is very large, reducing the expansion to zero. 

It is remarkable that the asymptotic value 0 = f in three dimensions (and correspondingly 0 = in two dimensions) was suggested many years ago by Flory.32 Using a mean field type of argument, and denoting by a the expansion factor over a chain with no excluded volume, Flory derived the equation (in three dimensions)... 

The reference state dimensions unperturbed dimensions, because of the unknown influence of the presence of crosslinks, possibly specific diluent effects, and perhaps at high swelling even an excluded volume effect. We have pointed out that (r2)0 may depend on the concentration of the diluent. Therefore the reference state is in general not a constant. We have also pointed out that, if (r2)0 contains a molecular expansion term due to an excluded volume effect, the use of the Flory-Huggins free enthalpy of dilution is no longer adequate. A difference between the % parameter in a network and the X parameter of the same polymer material but then in solution, may occur because the presence of crosslinks may modify %. 

This means that the only effect of the q factor on the [77]-M relationship is that of the excluded volume due to coil expansion in a good solvent. It follows from Equations 8 and 16 that... 

In this Section we consider several approaches which differ from the many-point density formalism discussed above. Szabo et al. [45] have introduced a novel method based on the density expansion for the survival probability, u>(t). Consider a system containing walkers (particles A) and N traps (quenchers B) in volume V in d-dimensional space. We assume that the particles have a finite size but the traps can be idealized as points and hence are ignorant of each other. When the concentration of the walkers is sufficiently low so that excluded volume interactions between them are negligible, one might focus on a single walker. 

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