Polymers internal motions

Light scattering spectra of random-coil polymers differ from spectra of colloidal particles random coils have observable internal modes. At small q, polymer and colloid internal modes involve distances small relative to so internal modes do not contribute to the time dependence of 5(, t). At large 5(, t) of a rigid particle reflects only center-of-mass motion, because rigid probe particles lack observable internal motions. In contrast, for large q internal modes of flexible molecules involve motions over distances comparable to and thus contribute directly to S q,t). Except at extreme dilution, interactions between polymer chains affect both polymer center-of-mass motion and polymer internal motions. 

Molecular motion in solids has been the object of many studies in the field of physical chemistry of polymers , but dynamic processes in molecular crystals of organic and inorganic compounds are less well investigated. In fact, the average chemist is not aware of the fact that processes like internal rotation or ring inversion proceed in solids quite often with barriers which are not very different from those found for these types of internal motion in the liquid state. Thus, for the equatorial axial ring inversion of fluorocyclohexane values of 42.4 and 43.9 kJ mol have been measured in the liquid and the solid, respectively. The familiar thermal ellipsoids of individual atoms obtained from X-ray studies are qualitative indicators of molecular motion in the crystal, but a more quantitative study of such processes is only possible after appropriate solid state NMR techniques are applied. 

Constraints may be introduced either into the classical mechanical equations of motion (i.e., Newton s or Hamilton s equations, or the corresponding inertial Langevin equations), which attempt to resolve the ballistic motion observed over short time scales, or into a theory of Brownian motion, which describes only the diffusive motion observed over longer time scales. We focus here on the latter case, in which constraints are introduced directly into the theory of Brownian motion, as described by either a diffusion equation or an inertialess stochastic differential equation. Although the analysis given here is phrased in quite general terms, it is motivated primarily by the use of constrained mechanical models to describe the dynamics of polymers in solution, for which the slowest internal motions are accurately described by a purely diffusive dynamical model. 

The spectrum of scattered light contains dynamical information related to translational and internal motions of polymer chains. In the self-beating mode, the intensity-intensity time correlation function can be expressed (ID) as... 

In the second half of this article, we discuss dynamic properties of stiff-chain liquid-crystalline polymers in solution. If the position and orientation of a stiff or semiflexible chain in a solution is specified by its center of mass and end-to-end vector, respectively, the translational and rotational motions of the whole chain can be described in terms of the time-dependent single-particle distribution function f(r, a t), where r and a are the position vector of the center of mass and the unit vector parallel to the end-to-end vector of the chain, respectively, and t is time, (a should be distinguished from the unit tangent vector to the chain contour appearing in the previous sections, except for rodlike polymers.) Since this distribution function cannot describe internal motions of the chain, our discussion below is restricted to such global chain dynamics as translational and rotational diffusion and zero-shear viscosity. 

Ferry and coworkers [118] extensively studied viscoelasticity of dilute solutions of stiff-chain polymers. Their results made clear that the stress or the storage and loss moduli for the solutions are sensitive to chain internal motions... 

The rotational diffusion coefficient Dr of a rodlike polymer in isotropic solutions can be measured by electric, flow, and magnetic birefringence, dynamic light scattering, and dielectric dispersion. However, if the polymer has some flexibility, its internal motion makes it difficult to extract Dr for the end-over-end rotation of the chain from data of these measurements. In other words, Dr can be measured only for nearly rodlike polymers. 

Since crosslinked polymer networks exhibit anisotropic internal motions, the effect of MAS on the line narrowing is explained assuming that the rigid lattice second moment AM2 consists of two terms, analogous to Eq. (34). Then, the residual line width in the NMR MAS experiment, Acomas, can be described by the following relation ... 

Heijboer [28] has reported the dynamic mechanical properties of poly(nethacrylate)s with different size of the saturated ring as side chain. The y relaxation in these polymers is attributed to a conformational transition in the saturated ring. In the case of poly(cyclohexyl methacrylate), the transition is between the two chair conformations in the cyclohexyl ring. However, this type of internal motion in hindered by rather high intramolecular barriers, which can reach about 11 kcal mol-1. 

One problem that arises is that although Newtonian mechanics is sufficient to describe the overall translational motion of particles of the size of atoms and molecules, quantum mechanics is required to describe their rotational and internal motion. Quantum mechanics is essential in dealing with the motion of particles as small as electrons. Because knowledge of quantum mechanics is not assumed of the reader of this book, we will be content to develop the framework into which quantum mechanical results can be later be inserted. We will apply this framework to two systems that can be treated classically, namely the monatomic ideal gas and polymer chains. 

Polymer Backbone Motion. Alternate descriptions of molecular motion utilize an effectively non-exponential autocorrelation function to describe polymer dynamics. One formalism is the use of a log-/2 distribution of correlation times in place of a single correlation time(14). Such a description may simulate the various time scales for overall and internal motions in polymers. 

In some polymers such segmental motions can be important, whereas in others (e.g., proteins) the overall skeleton is rigid, but there are rapid internal motions of moieties relative to the skeleton. In this case, relaxation and NOE data are often analyzed by the Lipari-Szabo formalism,96 which yields values for an overall correlation time rM, a correlation time for fast motions re, and a generalized order parameter S (see Eq. 7.16), which describes the amplitudes of the internal motions. 

Another approach is to assemble multiple chelates either covalently (11) (oligomer, polymer, and dendrimer) or non-covalently (12, 13) (micelle, liposome, and emulsion). These approaches all yield higher molecular relaxivities because of the assembly, but the per-ion relaxivity also is increased because motion is slowed. Fast internal motions can limit these relaxivity gains, but this limitation can be overcome by rigidifying the structure in some way (14). 

The Norrish type II reaction was found to occur with a quantum yield of 0.025 for solid films at room temperature [27], 0 appears to be independent of the presence of oxygen and of temperature above the glass transition temperature. Below Tg, however, the reaction is inhibited due to restriction of the freedom of internal motions of the polymer chains. In fact, the reaction has been shown to proceed by intramolecular hydrogen transfer, probably via a six-membered cyclic intermediate,... 

The existence of a bimodal linewidth distribution may be attributed to several factors. When a polymer is large, interference between segments of the same chain will give rise to an intramolecular scattering contribution to the linewidth. We have ruled out this possibility since K has a maximum value of 1.2 and is often much less than one in our experiments. Thus, our experiments cannot observe the contributions due to internal motions and they amount to, at most, one to two percent of the total scattered intensity.(lO) We have also made other studies whereby a second faster peak can be attributed to a pseudo-gel motion in semidilute solutions (l ). This explanation is unreasonable because the concentrations of our solutions are very small. We should not have reached the semidilute regime. 

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