Flory theory 2-dimensional chain

Consider a linear polymer chain with N monomers of length b, restricted to the air-water interface (two-dimensional conformations). Repeat the Flory theory calculation and demonstrate that the size R of the chain as a function of the excluded area a per monomer (two-dimensional analogue of excluded volume v) is... 

The theory of gelation (Flory, 1953,1974) has been summarized in Section 2.2.3. This theory regards gelation as the consequence of the random crosslinking of linear polymer chains to form an infinite three-dimensional network. The phenomenon is, of course, well illustrated by examples drawn from the gelation of polycarboxylic acids by metal ions. 

The Flory-Huggins theory begins with a model for the polymer solution that visualizes the solution as a three-dimensional lattice of TV sites of equal volume. Each lattice site is able to accommodate either one solvent molecule or one polymer segment since both of these are assumed to be of equal volume. The polymer chains are assumed to be monodisperse and to consist of n segments each. Thus, if the solution contains TV, solvent molecules and TV2 solute (polymer) molecules, the total number of lattice sites is given by... 

For t/j = 1 (linear chains). Equation (11.9a) provides the correct value, d = 2, corresponding to a macromolecular coil at the 0-point (see Table 11.2). As noted previously, d = 4/3 for a percolation cluster, irrespective of the dimension of the Euclidean space (see Table 11.1) therefore, from Equation (11.9a), we obtain df= 4, which is consistent with the Flory-Stockmayer theory [60] for phantom chains. For three-dimensional space, d > 3 has no physical meaning because the object cannot be packed more densely than an object having a Euclidean dimension. It is evident that this discrepancy is due to the phantom nature of the polymer chains postulated by Cates [56] it is therefore, necessary to take into account self-interactions of chains due to which the dimension of a polymer fractal assumes a value that has a physical meaning. 

This is the so-called Flory-Fisher scaling law (De Gennes 1979). The critical exponent v = 1 in (4.21) at the dimensionality d = 1 v = 3/4 at d = 2 v = 3/5 at d = 3 and v = 1/2 at d = 4. These critical exponents are consistent with that of self-avoiding walks obtained above from the computer simulations. The scaling law for the ideal chain model occurs only in 4D space of SAWs. In 3D space, the renormalization group theory yields the critical exponent as v = 0.588 0.001, which is in good consistency with the computer simulation results (Le Guillou and Zinn-Justin 1977). 

The Flory-Huggins theory uses the lattice model to arrange the polymer chains and solvents. We have looked at the lattice chain model in Section 1.4 for an excluded-volume chain. Figure 2.1 shows a two-dimensional version of the lattice model. The system consists of si,e sites. Each site can be occupied by either a monomer of the polymer or a solvent molecule (the monomer and the solvent molecule occupies the same volume). Double occupancy and vacancy are not allowed. A hnear polymer chain occupies N sites on a string of N-l bonds. There is no preference in the direction the next bond takes when a polymer chain is laid onto the lattice sites (flexible). Polymer chains consisting of N monomers are laid onto empty sites one by one until there are a total tip chains. Then, the unoccupied sites are filled with solvent molecules. The volume fraction of the polymer is related to rip by... 

The Flory model for AS of mixing a polymer chain with solvents has been influential in polymer chemistry for several decades. The model assumes the validity of the lattice theory to describe the change in the molecular configuration of the polymer in the presence of a solvent, just as it describes the patterns of the crystal stracture of molecules. The central point is the filling of lattice sites in a three-dimensional space by polymer segments and solvent molecules that is, how many ways can we fill up the lattice sites ... 

For very high concentrations of polymer solutions where the density fluctuations are not dominant, one can expect mean field arguments to have validity. The first theory to obtain the thermodynamics of polymer solutions is due to Flory and Huggins. In the Flory-Huggins theory each polymer chain is represented as a chain of n segments, each exactly equal in volume to a solvent molecule (which is comparable to F), A polymer solution may now be represented by a lattice divided into cells, each cell of which may be occupied either by a segment of the polymer molecule or by a solvent molecule. A two-dimensional representation of such a lattice is shown in Figure 8. 

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