Flory approximation

Two schemes may be used to obtain explicit expressions for F. One is a so-called Flory approximation where the terms of Eq. 1 are estimated for ideal uncorrelated chains. The corresponding free energy per chain, in units of the thermal energy kT, is ... 

It is interesting, for comparison, to apply the Flory approximation, parallel to Eq. 2, to a curved geometry. As an example, consider a star polymer with f arms and radius R. The segment volume fraction within the star can be written, tp ss fNa3/R3, so that FjtU/kT per arm can be written as v(fN)2 a3/fR3, giving rise to a total free energy per arm ... 

Here Jta(x) denotes the a-th component of the stationary vector x of the Markov chain with transition matrix Q whose elements depend on the monomer mixture composition in microreactor x according to formula (8). To have the set of Eq. (24) closed it is necessary to determine the dependence of x on X in the thermodynamic equilibrium, i.e. to solve the problem of equilibrium partitioning of monomers between microreactors and their environment. This thermodynamic problem has been solved within the framework of the mean-field Flory approximation [48] for copolymerization of any number of monomers and solvents. The dependencies xa=Fa(X)(a=l,...,m) found there in combination with Eqs. (24) constitute a closed set of dynamic equations whose solution permits the determination of the evolution of the composition of macroradical X(Z) with the growth of its length Z, as well as the corresponding change in the monomer mixture composition in the microreactor. 

The total free energy of a real chain in the Flory approximation is the sum of the energetic interaction and the entropic contributions ... 

The quality of solvent, reflected in the excluded volume v, enters only in the prefactor, but does not change the value of the scaling exponent u for any v > 0. The Flory approximation of the scaling exponent isu = 3/5 for a swollen linear polymer. For the ideal linear chain the exponent = 1/2. In the language of fractal objects, the fractal dimension of an ideal polymer is V — l/i/ = 2, while for a swollen chain it is lower T> — I/u = 5/3. More sophisticated theories lead to a more accurate estimate of the scaling exponent of the swollen linear chain in three dimensions ... 

Remember that the Flory approximation produced results which were much less exact (see Chapter 8, Section 2.1.1). 

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] ... 

The Eq. (22) gives unsatisfactory correspondence to numerical calculations for precise fiactals and the authors [51] expressed doubt about the general Flory approximation derivation possibility for fractal spaces (lattices), which would be both simple and exact at the same time. 

The authors [52] obtained the following Flory approximant for fractal spaces ... 

For the dimension d value estimation the authors [54, 55] used the simplest variant based on Flory approximation and proposed in Ref [56], For the dimension 4/3<[Pg.18]

Aharony, A. Harris, A. B. Flory approximant for self-avoinding walks on fractals. J. Stat. Phys., 1989, 54, (3/4), 1091-1097. 

Given the paucity of exact solutions in this area, it seems reasonable to look for some artificially constructed graphs, e.g. fractals, for which an exact solution can be found. This solution then, can be considered as an approximation to the original problem. The advantage of this approach, over other ad-hoc approximations like the Flory approximation, is that one is assured of weU-behavior requirements like the convexity of the free energy, and avoids problems like getting two different values for a quantity ( e.g. the pressure for hard-sphere systems), if one calculates it in two different ways within the same approximation. 

Once the distribution of polymer sizes is known, it is possible to dilute the sol, and to consider dilute solutions. Let us stress that the growth of the polymers is quenched before dilution and that the distribution function is given. Because of the excluded volume interactions the polymers swell and their fractal dimension changes from df to df. The new fractal dimension df may be obtained within a Flory approximation by considering a free energy similar to that in Eq. (6.27). The difference between a dilute solution and the reaction bath which was considered above is in the interaction term. We expect that the excluded volume interactions are present in the dilute case whereas they are fully screened in the previous case [40]. Therefore, this contribution has the same form as in relation (6.9) for linear chains. It is straightforward to minimize the free energy with respect to the radius, which yields. 

The large Z limit equation is the Flory approximation. The parameter 8 is independent of temperature and pressure. The closed packed volume rV of a molecule is independent of tanperature and pressure. The closed packed volume of a repeat unit is V. It is also the volnme of a lattice site. The closed packed volume of N, repeat oligomers (no holes) is... 

D. Lhuillier, A simple model for polymeric fractals in a good solvent and an improved version of the Flory approximation, J. Phys. France) 49, 705 (1988). 

The importance of the c parameter was first recognized by Prigogine et al. [70]. It is to be taken as a measure of the perturbation of internal rotations of the chain in addition to motions of the chain as a whole, in the dense medium. The Flory approximation [12, 13] is invoked to represent the combinatorial entropy... 

A theta solvent is a solvent in which the polymer will behave like an ideal chain. If a polymer is dissolved in a particular solvent at temperature and the temperature of the solution is decreased so T < 0, then the polymer coil will shrink. If the temperature is increased to T > 0, the polymer coil will expand to fill a larger volume. The competing effects of self-avoidance and the van Der Waals attraction have an impact on the density with which the polymer fills space, resulting in changes in the v exponent. The v exponent will only be equal to 0.5 in a theta solvent, but at high temperatures, we can use the Flory approximation to estimate the value of v for a self-avoiding chain. 

It is seen that the Flory approximation is inaccurate, while both other approximations describe the equation of state well at high volume fractions small volume fractions, however, neither of these approximations is very accurate, as expected, since in the dilute and semidilute concentration regime a scaling description of the equation of state is needed. 

Big. 7.23 Ratio between the Flory-Huggins critical temperature, Jf = N 1 — y)z(/ 2ks) and the actual critical temperature Tc for the self-avoiding walk model of polymer mixtures on the simple cubic lattice (Fig. 7.3) plotted versus concentration 1 - of sites taken by monomers (upper part) and versus the inverse square root of the chain length (lower part). Upper part refers to Af = 16 (for energy parameters coincide this is marked by a sohd dot). Curves are only drawn to guide the eye. Both the Flory approximation, eq. (7.34), which implies = 1, i.e., a horizontal straight line, and the Guggenheim... 

While the Flory approximation works well for some systems, this is not always the case. Two such examples are Sierpihski gaskets ind=2 and two-dimensional tethered membranes in = 3. For tethered membranes, the Flory-level theory predicts that df = 5/2 or d = D = 2 and d= 5. This state is commonly referred to as a crumpled membrane since the relaxed membrane should be isotropic. This expectation that the membrane would crumple was supported by renormalization group calculations and early MC simulations on small systems. However a number of more detailed and simulations found that the membranes... 

In most of the experimental studies, the copolymer chains are not so long relative to the homopolymers. Thus, mixing of the copolymer and homopolymer chains should be taken into account due to the penetration of homopolymers into the layer of chains anchored at the interface, whereas the copolymer chains can be either stretched (wet brush regime) or not (wet mushroom). Neglecting the composition gradients in the bmsh (Flory approximation), g, is given by [40, 75, 287] ... 

We shall use the dynamical scaling theory to describe the hydrodynamic properties of polymer solutions, focusing mainly on the expected universal behavior. We use a Flory approximation for the power law behavior, which turns out to be a much easier approach and allows a simple understanding of the important physical features often masked by a heavier formalism. For comparison with experiment we shall sometimes quote more detailed results obtained by renormalization group calculations. We will also discuss briefly the deviations from universal behavior related to crossover effects. 

To proceed it is necessary to obtain an explicit expression for Fchain- o simple approaches available are the Flory approximation and the blob picture. Within the Flory approximation the polymers are viewed as ideal, uncorrelated chains. Accordingly, the number of monomer-monomer interactions per site scales as Since the volume per chain is aL, Fchain/ v/a ) aL where vkT is the second virial coefficient. The elastic penalty of a Gaussian chain is F /kT iP /R where Ro is the unperturbed radius of the coil. Altogether... 

Both Fi f and F are overestimated within this approach because self avoidance correlations are neglected. The Flory approximation is nevertheless very useful (i) It typically yields the correct equilibrium dimensions because the two overestimates cancel when the equilibrium condition is invoked, (ii) It provides a useful upper bound for Fchain- (iii) Most important, the Flory approximation is simple to apply irrespective of difficulties such as complicated geometries. While it is a good idea to use the Flory approximation as a first step in the analysis of a problem, it is important to understand its limitations. Since the free energy density is overestimated, so is the osmotic pressure. As a result erroneous results are obtained for related properties such as the force law for brush compression. The overestimate of F j f and F i can result in wrong predictions for equilibrium dimensions when F hain includes extra terms and cancellation of errors may not be relied upon. Such is the case for aggregates of block copolymers where F hain must allow for a surface energy contribution. 

Both Fini and are smaller by a factor of in comparison to the expressions used in the Flory approximation. Pint is lower since the number of monomer-monomer contacts decreases because of the self avoidance. The extra factor in F / arises because the Gaussian chain consists now of blobs rather than monomers. Within this picture, the equilibrium state of the brush is specified by... 

Thus, Fchain as calculated blobologically is indeed lower than the Fchain obtained via the Flory approximation while the brush thickness obtained by the two methods is identical. 

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