Chain dimension

The subject of chain dimensions is concerned with relating the sizes and shapes of individual polymer molecules to their chemical structure, chain length and molecular environment. In the analysis of polymer solution thermodynamics given in the previous sections it was not necessary to 

The shape of a polymer molecule is to a large extent determined by the effects of its chemical structure upon chain stiffness. Since most polymer molecules have relatively flexible backbones, they tend to be highly coiled and can be represented as random coils. However, as the backbone becomes stiffer the chains begin to adopt a more elongated worm-like shape and ultimately become rod-like. The theories which follow are concerned only with the chain dimensions of linear flexible polymer molecules. More advanced texts should be consulted for treatments of worm-like and rod-like chains. 

The simplest measure of chain dimensions is the length of the chain along its backbone and is known as the contour length. For a chain of n backbone bonds each of length /, the contour length is nl. However, for linear flexible chains it is more usual, and more realistic, to consider the dimensions of the molecular coil in terms of the distance separating the chain ends, i.e. the end-to-end distance r (Fig. 3.2). 

This function is plotted in Fig. 3.4(b) and can be shown by simple differentiation (see problems) to have a maximum value at r = l/j8. It also is simple to show that lT(r) normalizes to unity, i.e. 

The mean square end-to-end distance, (r ), is the second moment of the radial distribution function and so is defined by the integral 

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In the limit that the number of effective particles along the polymer diverges but the contour length and chain dimensions are held constant, one obtains the Edwards model of a polymer solution [9, 30]. Polymers are represented by random walks that interact via zero-ranged binary interactions of strength v. The partition frmction of an isolated chain is given by... 

Forsman W.C., Effect of segment-segment association on chain dimension. Macromolecules, 15, 1032, 1982. 

Considering the chain dimensions, for very long gaussian chain, one predicts ... 

Equations (10) and (11) are characteristic of the Gaussian distribution, Eq. (8), irrespective of the relationship of 0 to chain dimensions in any given instance. In the particular case of the freely jointed chain assumes the value given by Eq. (6). Substituting Eq. (6) inEqs. (10) and (11) yields... 

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]. 

Figure 4.4 shows the histogram of R, which corresponds to the probability distribution function of the chain dimension. Information on the distribution was not available from the previous experiments in inverse space. The average radius of gyration, (i xy). was 138, 145, and 143 nm for the PMMA chains in thin films with thickness 15, 50, and 80 nm, respectively. The thickness of 15-80nm is relatively... 

Figure 4.4 Histogram of the lateral chain dimension for the PMMA-Pe chains in ultra-thin films with thickness 15, 50, and 80 nm.The PMMAchains with a molecularweightof4 x 10 were selected in the SNOM images and analyzed to construct the histogram. Reproduced with permission from The Society of Polymer Science, Japan.

The chain dimension in the height direction was evaluated as the thickness of the brush layer, I, relative to the chain contour length, io, by atomic force microscopy (AFM). Figure 4.10 shows the solvent dependence of the conformation of the PMMA brush. Whereas the brush chain changes its conformation in response to the solvent quality at the low graft density, the high-density PMMA brush does not show... 

Figure 4.10 Chain dimension of the PMMA brush in the normal direction to the substrate. L and Lq are the thickness ofthe brush layerand the contour length ofthe brush chain. The open and filled circles indicate the chain dimension in acetonitrile and benzene.

Reiter, J., Zifferer, G. and Olaj, O. F. (1989) Monte-Carlo studies of polymer-chain dimensions in the melt in 2 dimensions. Macromolecules, 22, 3120-3124. 

The ratios of mean-squared dimensions appearing in Equation (13) are microscopic quantities. To express the elastic free energy of a network in terms of the macroscopic (laboratory) state of deformation, an assumption has to be made to relate microscopic chain dimensions to macroscopic deformation. Their relation to macroscopic deformations imposed on the network has been a main area of research in the area of rubber-like elasticity. Several models have been proposed for this purpose, which are discussed in the following sections. Before that, however, we describe the macroscopic deformation, stress, and the modulus of a network. 

IR dichroism has also been particularly helpful in this regard. Of predominant interest is the orientation factor S=( 1/2)(3—1) (see Chapter 8), which can be obtained experimentally from the ratio of absorbances of a chosen peak parallel and perpendicular to the direction in which an elastomer is stretched [5,249]. One representation of such results is the effect of network chain length on the reduced orientation factor [S]=S/(72—2 1), where X is the elongation. A comparison is made among typical theoretical results in which the affine model assumes the chain dimensions to change linearly with the imposed macroscopic strain, and the phantom model allows for junction fluctuations that make the relationship nonlinear. The experimental results were found to be close to the phantom relationship. Combined techniques, such as Fourier-transform infrared (FTIR) spectroscopy combined with rheometry (see Chapter 8), are also of increasing interest [250]. 

Fluctuations are larger in networks of low functionality and they are unaffected by sample deformation. The mean squared chain dimensions in the principal directions are less anisotropic than in the macroscopic sample. This is the phantom network model. 

It is always easy to calculate idealized scattering curves for perfect networks. The experimental systems vary from the ideal to a greater or lesser degree. Accordingly, any estimate of the correctness of a theoretical analysis which is based on an interpretation of experiment must be put forth with caution since defects in the network may play a role in the physical properties being measured. This caveat applies to the SANS measurement of chain dimensions as well as to the more common determinations of stress-strain and swelling behavior. 

Approximately 5% of the chains were perdeuterated. The chain dimensions changed slightly or not at all upon crosslinking. 

A problem arises, in that the strong r b dependence of My requires that close overlap of spins be prevented. Thus, even though excluded volume interactions have no effect on chain dimensions in the bulk amorphous phase, it is important in the present application to build in an excluded volume effect (simulated with appropriate hard sphere potentials), so that occasional close encounters of the RIS phantom segments do not lead to unrealistically large values of M2. 

Figure 6 shows the importance of having separate exclusion distances for neighboring and non-neighboring fluorines. The neighbor fluorine exclusion is the main determinant of overall chain dimensions, serving to keep phenyl groups separated in the same manner as dyad statistics would have done. 

A second important characteristic is the value otj. of the elongation at which rupture occurs. The corresponding values of r/rm show that rupture generally occurred at approximately 80-90% of maximum chain extensibility (12). These quantitative results on chain dimensions are very important but may not apply directly to other networks, in which the chains could have very different configurational characteristics and in which the chain length distribution would presumably be quite different from the very unusual bimodal distribution intentionally produced in the present networks. 

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