Coils ideal chains

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. 

In good solvents, the mean force is of the repulsive type when the two polymer segments come to a close distance and the excluded volume is positive this tends to swell the polymer coil which deviates from the ideal chain behavior described previously by Eq. (1). Once the excluded volume effect is introduced into the model of a real polymer chain, an exact calculation becomes impossible and various schemes of simplification have been proposed. The excluded volume effect, first discussed by Kuhn [25], was calculated by Flory [24] and further refined by many different authors over the years [27]. The rigorous treatment, however, was only recently achieved, with the application of renormalization group theory. The renormalization group techniques have been developed to solve many-body problems in physics and chemistry. De Gennes was the first to point out that the same approach could be used to calculate the MW dependence of global properties... 

The lateral forces depend on temperature at high temperatures the repulsion interactions between particles prevail on the contrary, at low temperatures the attraction interactions prevail, so that there is a temperature at which the repulsion and attraction effects exactly compensate each other. This is the 0-temperature at which the second virial coefficient is equal to zero. It is convenient to consider the macromolecular coil at 0-temperature to be described by expressions for an ideal chain, those demonstrated in Sections 1.1-1.4. However, the old and more recent investigations (Grassberger and Hegger 1996 Yong et al. 1996) demonstrate that the last statement can only be a very convenient approximation. In fact, the concept of 0-temperature appears to be immensely more complex than the above picture (Flory 1953 Grossberg and Khokhlov 1994). 

Gaussian coils are characterized by a gaussian probability distribution [2] for the monomers and describe adequately flexible polymer blocks. Ideal chains follow random walk statistics, i.e.,... 

When soluble polymers are attached by one end to a surface, the thickness of the resulting layer, L, depends on the surface density of chains a as well as n and the excluded volume v/l3 (de Gennes, 1980). At low densities risolated chains extend 1/2 into the solution, creating a layer with the density profile shown in Fig. 26a and a thickness of L = n1/2f for ideal chains and L n3/sl in good solvents. When 1 the coils overlap and the interactions will cause the chains to expand away from the surface into the bulk. The configurations of the individual molecules and the density profile within the layer (Fig. 26b) differ markedly from the dilute situation. When cl2 1 the molecules become fully stretched. 

In the following two sections we first consider colls In solutions which are so dilute that coll-coll Interaction does not play a role. Section 5.2a deals with random-walk chains where the Interaction between the units within one chain may be neglected. We denote these as ideal chains. Section 5.2b treats swollen coils In which the monomeric units repel each other due to so-called excluded volume effects. 

In Chapter 2, we studied the conformations of an ideal chain that ignore interactions between monomers separated by many bonds along the chain. In this chapter we study the effect of these interactions on polymer conformations. To understand why these interactions are often important, we need to estimate the number of monomer monomer contacts within a single coil. This number depends on the probability for a given monomer to encounter any other monomer that is separated from it by many bonds along the polymer. 

A mean-field estimate of this probability can be made for the general case of an ideal chain in cf-dimensional space by replacing a chain with an ideal gas of N monomers in the pervaded volume of a coil R. The probability of a given monomer to contact any other monomer within this mean-field approximation is simply the overlap volume fraction (j>, of a chain inside its pervaded volume, determined as the product of the monomer volume and the number density of monomers in the pervaded volume of the coil NjR ... 

The polymers considered above have been uncharged. Another class of polymers are polyelectrolytes whose chains carry a fixed charge. For Debye lengths small in comparison with the chain size, polyelectrolytes take on a coiled structure. Despite the presence of charge, the description for a polyelectrolyte fluid is similar to that which is obtained for a neutral polymer corresponding to the ideal chain regime (de Gennes 1990). 

For the three-dimensional self-avoiding walks, the critical exponent of the polymer coil is 3/5, which is larger than the critical exponent of the ideal chain (1/2). This implies that the volume exclusion of the polymer chain leads to coil expansion. Such an expansion makes chain conformation deviate from its most probable state, causing a recovery force originated from the conformational entropy. Therefore, the single coil could not expand unlimitedly, and there exists a thermodynamic balance between the energy gain of volume exclusion and the entropy loss of chain conformation. 

Eq. (XI.41) can be understood easily when g > 1 (in which case the subunit is itself a small coil). The factor g conesponds to ideal chain behavior. The u correction is derived from the perturbation calculation described in eq. (1.42). The second correction kd results from a similar perturbation treatment applied to the three body interaction. The constants k and kt depend on the value chosen for g. 

Indeed the excess free energy in the globule state, P(R), involves two main contributions F = Fconf+Pmt- (The nearly ideal coil state is considered as the reference state with P= 0.) The conformational free energy F onf iS/ in fact, the minimal work required to confine the ideal chain in a sphere of radius R. To estimate it we represent the chain as a sequence of N/g ideal g-blobs such that each blob just fit in the sphere the blob size bgi/2 Obviously Pconf niust be proportional to the number of the blobs ... 

Chain conformations are nearly Gaussian (at length-scales X exceeding the persistence length, X 1), in particular, the coil size R = J ee follows the ideal chain law R -... 

Here Vo l/c is the volume per repeat unit (bead) and the ideal chain size R bN is used. Thus Uc 3> 1 for long chains. If a coil swells, more chains will come to overlap with it keeping the total density nearly constant. The swelling therefore does not affect the total energy of interactions... 

The second important solvent property is solvent quality with respect to the macromolecule. Depending on solvent quality, polymers adopt different conformations. In a theta solvent, interactions between monomers are canceled by interactions with the solvent, and the polymer thus behaves as an ideal chain with zero excluded volume. In good and athermal solvents, the polymer behaves as a real chain with positive excluded volume, that is, it adopts a more loosely packed coil conformation. In poor solvents, the excluded volume is negative and the chain is more compact than an ideal chain. Obviously, these differences in conformation may influence the mobility of spin labels and of spin probes interacting with the chain. Both solvent quality and viscosity depend on temperature. For example, many solvent-polymer pairs have a theta temperature at which ideal-chain behavior is adopted. 

Fig. 1 Schematic representation on the free-energy profile for a single stiff polymer, a = Rr/Ro, where Ro and R, are the gyration radii of ideal chain and real polymer chain, respectively. I Coil is stable, II Coexistence between coil and globule, and ELI globule is stable...

This is an empirical equation which accounts for the fact that for ideal chains, i.e. vanishing excluded volume interactions, the coil size in thermal equilibrium equals Rq j3 is a dimensionless coefficient of order unity. The first term gives the repulsion experienced on squeezing a polymer chain, the second term represents the retracting force built up on a coil expansion. As only the second term appears relevant for the case under discussion we ignore the first term and write... 

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. 

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