Conformations of an ideal chain

Polymers are random fractals, quite different from Koch curves and Sierpinski gaskets, which are examples of regular fractals. Consider, for example, a single conformation of an ideal chain, shown in Fig. 1.14. As will be discussed in detail in Chapter 2, the mean-square end-to-end distance of an ideal chain is proportional to its degree of polymerization. 

The conformation of an ideal chain, with no interactions between monomers, is the essential starting point of most models in polymer physics. In this sense, the role of the ideal chain is similar to the role of the harmonic oscillator or the hydrogen atom in other branches of physics. 

Every possible conformation of an ideal chain can be mapped onto a random walk. A particle making random steps defines a random walk. If the length of each step is constant and the direction of each step is independent of all previous steps, the trajectory of this random walk is one conforma-tion of a freely jointed chain. Hence, random walk statistics and ideal chain statistics are similar. 

One conformation of an ideal chain with one end at the origin and the other end within volume d/t di of position R. 

The stretching along the x axis, shown in Fig. 2.13, makes the stretched conformation of an ideal chain a directed random walk of tension blobs. This conformation is sequential in the x direction, but the y and z directions... 

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. 

Note that this entropic free energy alone has a minimum at i = Nb, which is the conformation of an ideal chain. 

Figure 1.10. Conformations of an ideal chain (a) and a real chain (b) in two dimensions.

In the semidilute solution, blobs do not feel the excluded volume by the other blobs. Therefore, a chain of blobs takes a conformation of an ideal chain consisting of A/gjv blobs of size Its radius of gyration in the semidilute solution at monomer density p is estimated as... 

We learned in Section 4.2.2 that the polymer chain in the sanidilute solution takes a conformation of an ideal chain. We therefore can use a random-walk model to construct the test chain. Let the random walk consist of N independent steps of step length b. Then, the contour length of the test chain is Nb, and the mean square end-to-end distance is Nb. ... 

The hard sphere model is based on the excluded volume of spherical particles. An excluded volume theory has been developed to account for the orientational ordering of liquid crystal molecules, assuming them to be hard rods. This is the Onsager theory and its variants, outlined in Section 5.5.2. Excluded volume interactions influence the conformation of polymer chains. The conformation of an ideal chain is described by a random walk. However,... 

Real chains in good solvents have the same universal features as self-avoiding walks on a lattice. These features are described by two "critical" exponents y and v. The first is related to chain entropy, the second to chain size a real chain has a size that is much larger than that of an ideal chain (Nv instead of N1/2, where v 3/5 in good solvents) in good solvents the conformation of the chain is "swollen". 

The second interaction can be considered as follows, for two monomer units separated by a large number of monomers, the probability of an overlap between the two is non-zero. If the balance of the interactions, monomer-solvent and monomer-monomer leads to a repulsive interaction, as happens in a good solvent, these two distant monomer units repel each other. This excluded volume interaction increases the volume occupied by the macromolecule compared to that of an ideal chain the chain is swollen in a good solvent. A new statistical description of tlie chain conformation is needed, but the key point is that the scaling law still applies and we have R=N, but with v = 0.588 instead of 0.5. The normalised scattering intensity in the asymptotic range becomes ... 

The simplest polymer lattice model is that of an ideal chain, that is, a chain of N connected bonds (N -I- 1 monomers) whose terminus is placed on the origin of a square lattice. Immediate chain reversals are not permitted, but no interaction energy is defined, which means that the chain can intersect and retrace itself (Figure 4). Although this model is unrealistic, its partition function can be easily calculated. It is a summation over all the different chain conformations thus, Zj = 4 X Each chain conformation is equally probable,... 

Because a random coil is the least structured conformation of an idealized polymer chain, it corresponds to the state of greatest entropy. Any stretching of the coil introduces order and reduces the entropy. Conversely, the formation of a random coil from a more extended form is a spontaneous process (provided enthalpy contributions do not interfere). The change in conformational entropy, the entropy arising from the arrangement of bonds, when a coil containing N bonds of length I is stretched or compressed by nl is... 

The idea here is that as the polymer is stretched to an increasing magnitude of R, there are a decreasing number of conformational states capable of producing that R value. The implied decreased disorder in the system means that the entropy of the chain decreases as the chain is stretched. The entropy of an ideal chain of size R is defined by the formula... 

Because the single macromolecules in the considered structures have a same periodical conformation, for a length of the chains which in the ideal case tends to infinity, the difference between the entropies of an ideally ordered structure and a disordered one is small. 

This expression accounts for the configurational entropy of an ideal binary mixture with identical molecular sizes, but not for that of a polymer solution, since polymer chains are large and flexible. For that case, more contributions arise from the chain conformational entropy, first considered by Meyer [19] and then derived by Huggins [20] and Flory [21]. In analogy with a nonreversing random walk on a lattice, the conformational contribution of polymer chains to the partition function is given by... 

V, is the molar volume of polymer or solvent, as appropriate, and the concentration is in mass per unit volume. It can be seen from Equation (2.42) that the interaction term changes with the square of the polymer concentration but more importantly for our discussion is the implications of the value of x- When x = 0.5 we are left with the van t Hoff expression which describes the osmotic pressure of an ideal polymer solution. A sol vent/temperature condition that yields this result is known as the 0-condition. For example, the 0-temperature for poly(styrene) in cyclohexane is 311.5 K. At this temperature, the poly(styrene) molecule is at its closest to a random coil configuration because its conformation is unperturbed by specific solvent effects. If x is greater than 0.5 we have a poor solvent for our polymer and the coil will collapse. At x values less than 0.5 we have the polymer in a good solvent and the conformation will be expanded in order to pack as many solvent molecules around each chain segment as possible. A 0-condition is often used when determining the molecular weight of a polymer by measurement of the concentration dependence of viscosity, for example, but solution polymers are invariably used in better than 0-conditions. 

In the simplest possible model for an Ideal chain, the bonds between atoms In the backbone are treated as vectors connecting volumeless points which do not Interact. Such a model chain is depicted in fig. 5.2 the (fluctuating) distance between the end points is denoted as r. If, moreover, any orientation between two consecutive bonds is assumed to have the same probability, the conformational properties of long chains can be described by the universal random-Jlight model, first introduced by Kuhn l. Let the chain have N randomly oriented bonds, each of length t. Such a model chain contains IV + 1 backbone atoms. When these bonds are assumed to be fully Independent of each other, the conformation resembles the trajectory of a particle diffusing under the action of a random force, for which the solution is well known -S- ). The mean square displacement [Pg.614]

FIGURE 4 Three of the most popular indirect models of the active site of the sweet taste receptor. (A) Main contour ofthe active site proposed by Temussi and coworkers (Kamphuis et al., 1992 Temussi et al., 1978,1984,1991), hosting a molecular model of aspartame in an extended conformation. (B) A topological model, developed by Goodman et al. (1987). The L -shaped model and an L -shaped conformation of aspartame are superimposed. The hydrophobic side chain of Phe is denoted X, since it corresponds to the Kier s dispersion point. (C) 3D model of an idealized sweetener proposed by Tinti and Nofre (1991). Besides the AH-B entity, the model has six additional interaction points connected by a complex network of distances. 

Excluded volume and solvent quality. Up till here, the volume taken up by the polymer itself, i.e., n times the volume of a monomer, has been neglected. In other words, such an ideal random chain has no volume, which would imply that two different segments can occupy the same place in the solvent at the same time. This is, of course, physically impossible, which is why the statistics of a real chain are different from those of a random walk (diffusion). Instead of this, a self-avoiding random walk should be considered, and the average conformation then is different, rm being proportional to n to the power 0.6, rather than 0.5. This means that the molecule is more expanded than an ideal chain. 

However, the fully extended conformation is only one of a great many it would be more meaningful to consider an average size of the macromolecule such as the mean square end-to-end distance, r2. As the name implies, the end-to-end distance is just the length of the vector connecting the two ends of the ideal chain. This average can be that for a given molecule at a number of times or that of an ensemble of identical molecules at the same time.5 Thus, for p chains that do not interact with one another,... 

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