Measures of Unperturbed Dimensions

Although theoretically sound, this method has proved to be difficult to impossible, in practice. In general, the light scattering method is hindered by a number of errors optical artifacts, molecular heterogeneity, and the errors inherent in the dual extrapolation required with respect to concentration and angle. In addition, exper- 

Viscosity measurements on polymers dissolved in theta solvents have proved more useful. However, again measurements are subject to difficulties due to poor solubility near the theta point. At the theta temperature the Flory-Fox relationship31 reduces to  

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The results of measurements of unperturbed dimensions for a series of stereoregular PMMA are reported. Comparison of the results obtained with those predicted by previous RIS models show good agreement. 

For the purpose of determination of the temperature coefficient for unperturbed dimensions of copolymer 3 (Table 1), [r ] values were measured in the same solvent (toluene) at different tempe-ratures. In accordance with the Shtockmayer-Fixman method, Kg=(/M)1/2xF0 values was deter-mined using the least-square technique. The temperature coefficient of unperturbed dimensions was calculated from the values obtained at different temperatures (Table 6) using the relation, suggested in the work [45] dinldT = 2/3x1 n A/ t//-. The coefficient of unperturbed coil dimension (dlnldl), determined for copolymer 3 (Table 1), equals 0.85xl0 3 deg"1 [44],... 

Comparison of Chain Dimensions. Knowledge of the intrinsic viscosities and molecular weights of a series of PMDA/DAPE polyamic acids allows estimation of the unperturbed chain dimensions. Comparison may then be made with the cured polyimide, with results obtained by other workers, and with calculated values. The expressions in the literature using values for [n] obtained in good solvents generally involve extrapolation of [t)]/M toM = 0, where excluded volume effects are presumed to be minimal (12-14). Although this method is not strictly valid, it is useful for comparison purposes when direct measurement of the dimensions is impractical. The data in Table I for the polyamic acid in distilled NMP and in the poorer mixed solvent NMP/dioxane yield (see Figure... 

There are two basic ways in which measurements of the unperturbed dimensions are obtained (1) determination of unperturbed dimensions directly, by measurements in theta solvents and (2) determination of the perturbed dimensions in a good solvent and extrapolation of the values to the unperturbed state using one of the existing theories. Both methods have been widely used as will be shown. 

Determination of the unperturbed dimensions of the polymeric pro-cyanidins presents special challenges. Well-characterized samples of high molecular weight are not readily available, and the solubility characteristics of the high polymers are incompatible with the application of the classical techniques for the measurement of the dimensions of a macromolecule. An alternative route to the unperturbed dimensions exploits structural determinations in the solid state (8, 9), spectroscopic studies of well-characterized oligomers in dilute solution (iO, ii), molecular mechanics (MM2 software) (i2) calculations (i3, 14), and rotational isomeric state analysis (15-17) to provide a realistic description of the dimensions of the high polymers. A vital piece of information comes from the time-resolved fluorescence of the monomers and oligomers of well-defined covalent structure. The fluorescence measurements also show promise for the characterization of the com-... 

From the above considerations it is clear that measurements of chain dimensions and hydrodynamic properties of an isolated, unperturbed ring polymer... 

The mean dimensions of the macromolecular coils in the entangled system are found to approach their unperturbed values, i.e. values they would have in a 0-solvent. The coil dimensions in the concentrated system are the same as the dimensions of ideal coils. This is confirmed by direct measurements of the dimensions of macromolecular coils in concentrated solutions and melts by neutron scattering [30, 31]. 

Theta temperature (Flory temperature or ideal temperature) is the temperature at which, for a given polymer-solvent pair, the polymer exists in its unperturbed dimensions. The theta temperature, , can be determined by colligative property measurements, by determining the second virial coefficient. At theta temperature the second virial coefficient becomes zero. More rapid methods use turbidity and cloud point temperature measurements. In this method, the linearity of the reciprocal cloud point temperature (l/Tcp) against the logarithm of the polymer volume fraction (( )) is observed. Extrapolation to log ( ) = 0 gives the reciprocal theta temperature (Guner and Kara 1998). 

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. 

Deb P.C., Chatterjee S.R. Unperturbed dimension of polymer molecules from viscosity measurements. Die Makromolekulare Chemie 120,1 (1968) 49-57. 

The fractal dimension measures how open or packed a structure is lower fractal dimensions indicate a more open system, while higher fractal dimensions indicate a more packed system (22). Theories relating the fractal dimension to the relaxation exponent, n, have been put forward and these are based on whether the excluded volume of the polymer chains is screened or unscreened under conditions near the gd point (23). It is known that the excluded volume of a polymer chain is progressively screened as its concentration is increased, the size of the chain eventually approaching its unperturbed dimensions. Such screening is expected to occur near the... 

The mean-square dipole moments of POE and POMg are determined from dielectric constant measurements on dilute solutions in benzene. The values obtained are in good agreement with those predicted using the RIS models for these chains. In addition, the unperturbed dimensions of POMg are calculated as a function of molecular weight using the RIS theory. 

The unperturbed dimensions and their temperature coefficients are evaluated for poly n-pentene-1), poly(n-butene-l), and PS with the RIS model. The calculated values of the unperturbed dimensions for atactic and isotactic chains are in good agreement with the experimental data. The measured temperature coefficients, however, are described satisfactorily by the model for atactic polypentene and polybutene only. 

The unperturbed chain dimensions of near-monodisperse atactic PS are evaluated from intrinsic viscosity measurements. Negative values for the temperature coefficient of chain dimensions are found. Under conditions where specific solvent effects are eliminated or minimized, measurements yield results in excellent agreement with the theoretical predictions for atactic PS. 

Values of the dipole moment ratio of PNS are obtained from dielectric measurements. From thermoelastic experiments, performed on polymer networks, the temperature coefficient of the unperturbed dimensions is determined. Analysis of these results using the RIS model is performed leading to the parameters given above. 

Values of the mean-square dipole moment of PNA are determined from measurements of dielectric constants and refractive indices of the polymer in benzene. The dipole moment ratio and the temperature coefficient of both the dipole moment and the unperturbed dimensions are critically interpreted using the RIS model. Good agreement between theory and experiment is obtained by assuming that the gauche states about C(CH3)2— CH2 bonds have an energy 2.5 kJ mol-1 lower than the alternative trans states. 

The dipole moment ratio and the temperature coefficient of both the dipole moment and the unperturbed dimensions of the polyesters PDA and PDS are measured. The experimental value of dlln 0) / d Tshows an anomalous dependence on the elongation ratio of the networks at which the thermoelastic measurements are performed. Although the rotational states scheme gives a fairly good account of the polarity of the chains, it fails in reproducing the experimental values of d (In 0) / d T, the causes of this disagreement are discussed. 

More quantitative chemical evidence for random coil configuration comes from cyclization equilibria in chain molecules (49). According to the random coil model there must be a very definite relationship among the concentrations of x-mer rings in an equilibrated system, since the cyclization equilibrium constant Kx should depend on configurational entropy and therefore on equilibrium chain and ring dimensions. Values of /Af deduced from experimental values on Kx for polydimethylsiloxane, both in bulk and in concentrated solution, agree very well with unperturbed dimensions deduced from dilute solution measurements(49). 

This conclusion permits comparison of the thermomechanical and thermoelastic results for various networks. The most reliable data are summarized in Table 2. The temperature coefficients of the unperturbed dimensions of chains d In intermolecular interactions of the configuration of the network chains. 

It is well established that the excluded volume effect vanishes under a special condition of temperature or solvent, which is usually known as the Flory theta temperature or solvent. Thus, light scattering measurements performed on solutions under theta conditions can furnish direct knowledge of the unperturbed dimensions [see, for example, Outer, Carr and Zimm (207) Shultz (233) and Notley and Debye (207)]. Viscosity measurements, though less directly, can also furnish similar knowledge with the aid of the Flory-Fox equation (103,109), which may be written... 

According to the statistical-mechanical theory of rubber elasticity, it is possible to obtain the temperature coefficient of the unperturbed dimensions, d InsjdT, from measurements of elastic moduli as a function of temperature for lightly cross-linked amorphous networks [Volken-stein and Ptitsyn (258 ) Flory, Hoeve and Ciferri (103a)]. This possibility, which rests on the reasonable assumption that the chains in undiluted amorphous polymer have essentially their unperturbed mean dimensions [see Flory (5)j, has been realized experimentally for polyethylene, polyisobutylene, natural rubber and poly(dimethylsiloxane) [Ciferri, Hoeve and Flory (66") and Ciferri (66 )] and the results have been confirmed by observations of intrinsic viscosities in athermal (but not theta ) solvents for polyethylene and poly(dimethylsiloxane). In all these cases, the derivative d In sjdT is no greater than about 10-3 per degree, and is actually positive for natural rubber and for the siloxane polymer. 

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