The Unperturbed Polymer Chain

The quantitative treatment of a will be given in Chapter XIV. We note here merely that it often exceeds unity appreciably. Hence the differentiation is important. 

Perturbed and unperturbed polymer dimensions deduced from intrinsic viscosity measurements, according to procedures which will be discussed later, are given in Table XXXIX of Chapter XIV. The 

It may be shown that when the polymer concentration is large, the perturbation tends to be less. In particular, in a bulk polymer containing no diluent a = l for the molecules of the polymer. Thus the distortion of the molecular configuration by intramolecular interactions is a problem which is of concern primarily in dilute solutions. In the treatment of rubber elasticity—predominantly a bulk polymer problem—given in the following chapter, therefore, the subscripts may be omitted without ambiguity. 

Derivation of the Gaussian Distribution for a Random Chain in One Dimension.—We derive here the probability that the vector connecting the ends of a chain comprising n freely jointed bonds has a component x along an arbitrary direction chosen as the x-axis. As has been pointed out in the text of this chapter, the problem can be reduced to the calculation of the probability of a displacement of x in a random walk of n steps in one dimension, each step consisting of a displacement equal in magnitude to the root-mean-square projection l/y/Z of a bond on the a -axis. Then 

Introducing Stirling s approximations, n /je for the factorials, which is legitimate provided both n and n m are large, and consolidating the resulting expression, we obtain 

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The particularly simple relationships between the average end-to-end distance of the random coil and the chain length that are derived in section 2.4 are valid under the ideal solution conditions referred to as theta conditions. The dimension of the unperturbed polymer chain is only determined by the short-range effects and the chain behaves as a phantom chain that can intersect or cross itself freely. It is important to note that these conditions are also met in the pure polymer melt, as was first suggested by Flory (the so-called Flory theorem) and as was later experimentally confirmed by small-angle neutron scattering. 

The inter- and Intramolecular contributions to the entropy and energy of fusion are calculated for several linear aliphatic polyesters and polyamides assuming the fusion process consists of two Independent contributions the volume expansion (intermolecular contribution) and the increase in the conformational freedom of each polymer chain on melting (intramolecular contribution). The intramolecular entropy and energy contributions are obtained from the configurational partition function and Its temperature coefficient calculated for an isolated, unperturbed polymer chain using the RIS approximation. 

Two conflicting theoretical views concerning the flexibility of polymer chains and the role of the volume effect and the draining effect on fry] are discussed in the literature polymer chains of typical flexibility such as vinyl polymer chains, and a large value of Ip] can be interpreted in terms of the excluded volume effect (view point A) polymer chains are semi- or inflexible and their large unperturbed chain dimension is mainly responsible for a large [ry] (view point B). The former has its foundation on the two parameter theory 110. Untill 1977 these inconsistencies constituted one of the most outstanding problems yet unsolved in the science of polymer solutions. 

Under theta conditions the polymer coil is not expanded (or contracted) by the solvent and is said to be in its unperturbed state. The radius of gyration of such a macromolecule is shown in Section 4.4.1 to be proportional to the square root of the number of bonds in the main polymer chain. That is to say, if M is the polymer molecular weight and A/q is the formula weight of its repeating unit, then... 

A complication arising from the extension of the theory to flexible macromolecules is that in general, the intermolecular and intramolecular radial distribution functions depend on each other.In modeling the bulk of a one-phase polymer melt, however, the situation resolves itself because the excluded volume effect is insignificant under these conditions the polymer chains assume unperturbed dimensions (see also the section on Monte Carlo simulations by Corradini, as described originally in Ref. 99). One may therefore calculate the structure of the unperturbed single chain and employ the result as input to the PRISM theory to calculate the intermolecular correlation functions in the melt. 

The chain-end functionalized polymer is a very attractive material that possesses an unperturbed polymer chain with desirable physical properties (such as melting temperature, crystallinity, glass transition temperature, melt flow, etc.) that are almost the same as those of the pure polymer. Nevertheless, the terminal reactive group at the polymer chain end has good mobility and can provide a reactive site for many applications. This includes adhesion to the substrates, reactive blending, and formation of block copolymers. 

Hory and Fox suggested that as the viscosity of a polymer solution will depend on the volume occupied by the polymer chain, it should be feasible to relate coil size and [rj]. They assumed that if the unperturbed polymer is approximated by a hydrodynamic sphere, then [Tjjg, the limiting viscosity number in a theta solvent, could be related to the square root of the molar mass by... 

Chain stiffness and the effects of excluded volume became the dominating issue in the years between 1980 and the start of the new millennium. Percolation simulations indicated strong effects on the unperturbed polymer conformations due to excluded volume interactions [4]. With specially synthesized model substances (prepared by the Burchard group), the transition from mean-field to highly perturbed conformation was explored [5-17]. Studies in 1996 [8] on randomly branched, and in 2004 on hyperbranched polymers [8, 18-20], showed that the fractal conception could be quantitatively adjusted to the scattering behavior of linear and branched structures over the whole (/-domain and offered valuable insight into the structure in space [16]. 

It has been experimentally shown that the second moment of the end-to-end distance unperturbed polymer chains, which only appear under so-called theta conditions, is proportional to the... 

The phantom (unperturbed) polymer chain can be represented by a hypothetical chain with n = n/C freely jointed segments each of length / = Cl If n and / are replaced by n and / in the equation for the freely jointed chains, eq. (2.95) is obtained. 

Obviously, the temperature dependent parameter indicates the difference between the real chain and rwm structure. A review of the Cpf values for a great number of polymers can be found in reference [11]. The unperturbed real chain exhibits swelled conformations (in contrast to the rwm approximation) due to the iutra-molecular short-range interactions and almost fixed bond angles. This consequently leads to the temperature dependent end-to-end distance and other peculiarities such as the non-zero energy term in the rubber elastic response of real polymers [12]. 

Figure 4 (curve 1) shows that in the absence of extension the distribution function W(fi) lies in the range 0 < /S < 0.2 for relatively long chains. In other words, in the absence of external forces, crystallization of flexible-chain polymers always proceeds with the formation of FCC since in the unperturbed melt the values of /3 are lower than /3cr. For short chains, the function W(/3) is broader (at the same structural flexibility f) (Fig. 4, curve 2) and the chains are characterized by the values of > /3cr, i.e. they can crystallize with the formation of ECC. Hence, at the same crystallization temperature, a... 

Theta temperature is one of the most important thermodynamic parameters of polymer solutions. At theta temperature, the long-range interactions vanish, segmental interactions become more effective and the polymer chains assume their unperturbed dimensions. It can be determined by light scattering and osmotic pressure measurements. These techniques are based on the fact that the second virial coefficient, A2, becomes zero at the theta conditions. 

The conformation of polymer chains in an ultra-thin film has been an attractive subject in the field of polymer physics. The chain conformation has been extensively discussed theoretically and experimentally [6-11] however, the experimental technique to study an ultra-thin film is limited because it is difficult to obtain a signal from a specimen due to the low sample volume. The conformation of polymer chains in an ultra-thin film has been examined by small angle neutron scattering (SANS), and contradictory results have been reported. With decreasing film thickness, the radius of gyration, Rg, parallel to the film plane increases when the thickness is less than the unperturbed chain dimension in the bulk state [12-14]. On the other hand, Jones et al. reported that a polystyrene chain in an ultra-thin film takes a Gaussian conformation with a similar in-plane Rg to that in the bulk state [15, 16]. 

Isolated unperturbed polyoxyethylene chains have been simulated on the 2nnd lattice [154], The literature contains RIS models for a large number of polyethers [124], and it is likely that most of these chains could be mapped onto the 2nnd lattice with little difficulty. It is also likely that the work on PP [156,158] can be extended to other vinyl polymers, such as poly(vinyl chloride). This capability should permit the construction and complete equilibration of amorphous poly(vinyl chloride) cells larger than those described to date. They may be large enough to address issues arising from the weak crystallization reported for these systems [174]. 

Let us consider a polymer chain with N->oo identical skeletal atoms, either in solution or in the melt, representing our polymer system. Our reference temperature is T0, i.e., the temperature above which no bundles may effectively contribute to crystallization. At T = T0 the chain is assumed to be unperturbed and its configurational partition function is ZN(T0) = kN (N -> oo) [107] for simplicity we use a reduced form Zn = Z /kN (henceforth simply the partition function) so that Zn(T0) = 1. Only at T < To effective bundles may form, see Fig. 1, and we have ZN(T) = 1 + AZN(T - T0) note that the unit term corresponds to the bundle-free infinite-chain configuration. Each bundle with n chain atoms in -c N) will contribute to AZn... 

Here, a is the elongation ratio of the polymer chains in any direction and (r/j1/2 is the root-mean-square, unperturbed, end-to-end distance of the polymer chains between two neighboring crosslinks (Canal and Peppas, 1989). For isotropically swollen hydrogel, the elongation ratio, a, can be related to the swollen polymer volume fraction, u2,j> using Eq. (11). 

The unperturbed end-to-end distance of the polymer chain between two adjacent crosslinks can be calculated using Eq. (12) where Cn is the Flory... 

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