Chains in a Good Solvent

The expansion must be such that each site (i) is associated with a second moment Sf . Thus, odd powers of H drop out, and we are left with graphs proportional to HItY p = 1, 2...). In one such graph, we havep selfavoiding (and mutually avoiding) chains of lengths Ni... N . The total number of links are = Ni +. .. + Np. An example with p = 2 is shown in Fig. X.S. 

Note that in this system the chains are polydisperse, but their average degree of polymerization N will be determined by a suitable choice of K and r. Let us introduce a monomer concentration (per site) 1 and a polymer concentration (number of chains divided by the number of sites) Using the standard relations between concentrations and fiigatities, we have 

A typical thermodynamic quantity to be derived from the calculation is the osmotic pressure 11 (divided by the polymer temperature 7). A general thermodynamic theorem relates II to the grand partition function 

When this is minimized with respect to M we get an implicit equation for the equilibrium magnetization  

If we insert the resulting value of M = M(/f) into F(M), we reach the conjugate potential G H) = It is this potential which is related 

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Polymer chains at low concentrations in good solvents adopt more expanded confonnations tlian ideal Gaussian chains because of tire excluded-volume effects. A suitable description of expanded chains in a good solvent is provided by tire self-avoiding random walk model. Flory 1151 showed, using a mean field approximation, that tire root mean square of tire end-to-end distance of an expanded chain scales as... 

The large-scale structure of polymer chains in a good solvent is that of a self-avoiding random walk (SAW), but in melts it is that of a random walk (RW).11 The large-scale structure of these mathematical models, however, is... 

This function also gives an accurate description of the behavior of a linear chain in a good solvent (the expansion of the chain size is scaled by the x variable) except for very high values of x, corresponding to short distances between units. These short distances are dominated by the correlation hole effect due to EV [16,26]. 

The osmotic second virial coefficient A2 is another interesting solution property, whose value should be zero at the theta point. It can be directly related with the molecular second virial coefficient, expressed as B2=A2M /N2 (in volume units). For an EV chain in a good solvent, the second virial coefficient should be proportional to the chain volume and therefore scales proportionally to the cube of the mean size [ 16]. It can, therefore, be expressed in terms of a dimensionless interpenetration factor that is defined as... 

In good solvents we thus have B°N = R, where R is either the radius of the mushroom (roughly equal to the radius of gyration In solution) or of the pancake. In the further analysis, we concentrate on the pancakes. From computer simulations abd scaling arguments ) it is known that for a two-dimensional chciin R. This exponent is between that for a three-dimensional chain in a good solvent (R - N ) and a one-dimensional excluded-volume chain (which is a rod... 

Sikorski and Romiszowski455 study confined branched star polymers by on-lattice MC simulation. Attractive forces are excluded and only excluded volume accounted for, thus making the simulations relevant for chains in a good solvent. Contrary to expectation, they find that the diffusion constant is very similar for either moderate or highly confined chains and scales approximately as A 1, though a more accurate representation is suggested by... 

If neutral and charged polymer brushes are exposed to solvents, a very interesting and rich phase behavior can be observed. As a consequence, such surface-attached brushes have become the focus of considerable theoretical efforts in studies on the structure and phase behavior of such polymer chains in contact with a solvent. The thickness of a neutral brush scales linearly with the degree of polymerization, N, which is in stark contrast to the well-known characteristics of free polymer chains in a good solvent, where the radius of the coils scales as R oc AT0 58. For neutral brushes the simple scaling... 

The conformation of a single chain in a good solvent (left side) is a self-avoiding walk of thermal blobs while the conformation in a poor solvent (right side) is a collapsed globule of thermal blobs. 

The free energy of stretching a real linear chain in a good solvent has a stronger dependence on size R than the quadratic dependence of the ideal chain ... 

Schematic representation of a chain in semidilute solution in a good solvent. The plot shows the scaling of the end-to-end distance r with the number of monomers n in a subsection of a chain in a good solvent on logarithmic scales.

Using Eq. (3.77) for the size of the chain in a good solvent with intermediate excluded volume v in Eq. (8.25), and combining with the -solvent result of Eq. (8.25) with v= j2, yields a general expression for the Zimm time in dilute polymer solutions ... 

As will be seen later on, the renormalization theory provides a solution to these difficulties but unfortunately at the cost of complications which, until now, explicitly prevented the construction of a reasonably realistic model of a chain in a good solvent. 

As we saw in the preceding sections, the theory of isolated chains in a good solvent faced many difficulties and it is clear that, in principle, it is even more difficult to work out a theory of polymer solutions at finite concentrations. 

In order to determine the swelling of a polymer chain in a good solvent, we have to measure its size and to compare it with the size of the equivalent Brownian chain, i.e. the size that the chain would have in the Brownian state. This Brownian state is, in fact, only a concept however, by setting the sample in an adequate environment, one can get a situation close to this concept. 

The radius R is dependent on the solvent. Where the polymer is represented as an ideal, freely jointed chain, no account is taken of the effect of the solvent. This represents the borderline case between a poor and good solvent. Here, each link is in a random direction with respect to its neighbors and the end-to-end distance is proportional to aN where a is the effective monomer length. This distance is much smaller than the molecular length aN and is characteristic of the molecular radius R. For the more common case of a chain in a good solvent, excluded volume effects must be considered, and where this volume is of the order of it can be shown that (de Gennes 1979, 1990)... 

The box-like cell model of a PE star can be considered as a generalization of a classical mean-field Flory approach, which was first suggested to describe the swelling of a polymer chain in a good solvent [90], The Flory approach estimates the equilibrium dimensions of a macromolecule, as a function of its parameters, by balancing the free energy of intramolecular (repulsive) interactions with the conformational entropy loss of a swollen chain. Within the box-like approximation, the star is characterized by the radius of its corona, R (end-to-end distanee of the arms), or by the average intramolecular concentration of its monomers ... 

Here Cp and C are numerical prefactors /b and /a the effective segment lengths of insoluble and soluble blocks and Aa denote the respective number of repeat units and v is the excluded volume parameter controlling the conformation of the chain. In a good solvent, v takes the well-known value 0.588. 

The Real Chain in a Good Solvent - Floiy s Approximation.37... 

THE REAL CHAIN IN A GOOD SOLVENT - FLORY S APPROXIMATION... 

Comparison of Equations (5) and (36) shows that the size of chains in a good solvent is greater than that of perfect chains, R > R. It means chains are swollen in a good solvent. 

Formally, this transition is very similar to the worming transition. Flory s approach to a real chain in a good solvent is the base of discnssion (see paragraph 3.6). Flory considered in his seminal paper the balance between attraction among the monomers, wanting to collapse the polymer chain into itself, and the entropy of the chain, wanting to expand the chain. Flory s approach revealed the end-to-end distance of the swollen chain in a good solvent... 

Dilute polymer chains in a good solvent are swollea... 

The important feature of this expression is the fact that the brush height depends on iV in a linear way, rather than as the square root, as for an ideal chain, or the 3/5 power, as it does for an isolated chain in a good solvent. This means that the chains are strongly stretched this is most easily seen by rewriting (6.1.5) in terms of the Flory radius of gyration of an isolated chain Rp and the average distance between grafting points D. This yields... 

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